Waveguide Qed in the Squeezed Vacuum

WAVEGUIDE QED IN THE SQUEEZED VACUUM

A Dissertation by JIEYU YOU

Submitted to the Ofﬁce of Graduate and Professional Studies of Texas A&M University in partial fulﬁllment of the requirements for the degree of DOCTOR OF PHILOSOPHY

Chair of Committee, M. Suhail Zubairy Committee Members, Alexei Sokolov Aleksei Zheltikov Philip R. Hemmer Head of Department, Grigory Rogachev

May 2020

Major Subject: Physics

The well-known Purcell effect shows that the spontaneous decay rate of an emitter can be affected by the electromagnetic environment with which the emitters interact. One of the most famous and popular examples is the squeezed vacuum. Although the squeezed vacuum does not change the density of states of the electromagnetic modes, it can modify the decay rate as well as the dephasing rate of the emitters. The interaction between a single atom and the squeezed vacuum has been widely studied, while only a few publications deal with the multiple-atom system. Despite the fact that the dipole-dipole interaction induced by ordinary vacuum depends on the relative separation of atoms, there are only a few papers studying the impact of atomic separation in the squeezed vacuum. In this dissertation, we show that the interaction induced by the squeezed vacuum depends on the center of mass positions of the atoms, which is essentially different from that in the ordinary vacuum. We also illustrate how to choose the coordinate system to make the center of mass position reasonable and well-defined. Although the squeezed vacuum theory has been widely studied, it is impractical to generate a broadband squeezed vacuum reservoir which squeezes all modes in the 3-dimensional (3D) space. Recently, photon transport in a one-dimensional (1D) waveguide coupled to quantum emitters (well known as "waveguide-QED") has attracted much attention due to its possible applications in quantum device and quantum information. In contrast to the 3D case, squeezing in 1D is more experimentally feasible. Suppression of the spontaneous decay rate and the linewidth of the resonance fluorescence atom has been experimentally demonstrated in a 1D microwave transmission line coupled to a single artificial a t om. However many-body interaction in a 1D waveguide QED

system coupled to the squeezed vacuum has still not yet been studied. In this dissertation, we apply our theory to the 1D waveguide-QED system with the squeezed reservoir. Contrary to the traditional result that the dephasing rate of a single atom is a constant, our calculation shows that the dephasing rate is actually position-dependent. As the dipole-dipole interaction is involved in the atomic system, both the atomic separation and center of mass position have impacts on the decay ii rate, dephasing rate, and the emitted resonance ﬂuorescence spectrum. Moreover, the stationary maximum entangled NOON state can be achieved if atomic transition frequency is resonant with the center frequency of the squeezed vacuum. In light of the fact that two qubits can be treated as a whole to be a four level atomic system, we also study the dynamics of Ξ-type atoms driven by a squeezed vacuum. We get the interesting result that the atomic system’s steady state is a pure state, and a complete population inversion can occur when the coupling between the atomic dipole and the squeezed vacuum satisfy some certain conditions. We also mathematically prove that the steady state of a many-body system is nothing else but the direct product of that in the single atom case even when dipole-dipole interaction is involved.

iii DEDICATION

To my mother, Aihua Hu, and my father, Sijian You.

iv ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. M. Suhail Zubairy, for giving me the opportunity to learn from him. He is so patient, erudite, and full of knowledge that I could not have ﬁnished this dissertation without his instruction and guidance. I also want to thank Dr. Marlan O. Scully, the leader of IQSE (The Institute for Quantum Science and Engineering) for supporting me with HEEP fellowship and inviting me to Wyoming summer school, where I could learn from many world famous physicists. Thanks also to my colleagues Dr. Zeyang Liao, Dr. Xiaodong Zeng, Dr. Shengwen Li, Dr. Phililp R. Hemmer, Dr. Girish Agarwal, and Dr. Fuli Li for their collaboration. Their help was essential in many of my projects. It has been my great pleasure to work with and learn from them. I also want to extend my gratitude to the department faculty who taught me physics courses and the staff who helped me prepare for my defense.

v CONTRIBUTORS AND FUNDING SOURCES

Contributors

This work was supported by a dissertation committee consisting of Professor M. Suhail Zubairy, Alexei Sokolov, and Aleksei Zheltikov of the Department of Physics and Atronomy and Professor Philip R. Hemmer of the Department of Electrical & Computer Engineering. All work conducted for the dissertation was completed by the student independently.

Funding Sources

Graduate study was supported HEEP fellowship, a grant from the Qatar National Research Fund (QNRF) under NPRP project 8-352-1-074, and a grant from King Abdulaziz City for Science and Technology(KACST).

vi NOMENCLATURE

IQSE The Institute for Quantum Science and Engineering

TAMU Texas A&M University

QED Quantum electrodynamics

3D three dimensional

TE modes traverse electric modes

TM modes traverse magnetic modes

vii TABLE OF CONTENTS

Page

ABSTRACT ...... ii

DEDICATION...... iv

ACKNOWLEDGMENTS ...... v

CONTRIBUTORS AND FUNDING SOURCES ...... vi

NOMENCLATURE ...... vii

TABLE OF CONTENTS ...... viii

LIST OF FIGURES ...... x

1. INTRODUCTION...... 1

2. SQUEEZED VACUUM RESERVOIR AND TRADITIONAL RESERVOIR THEORY ... 4

2.1 Introduction to the squeezed vacuum ...... 4 2.2 Traditional reservoir theory ...... 14

3. A MODIFIED RESERVOIR THEORY IN THE SQUEEZED VACUUM...... 20

3.1 A new master equation from a modiﬁed mode function ...... 20 3.2 Master equation in the quasi-one-dimensional waveguide ...... 31 3.3 Dynamics of a single qubit ...... 40 3.4 Dynamics of multiple qubits coupled by the dipole-dipole interaction...... 41 3.5 Generalization to multi-level atoms ...... 43

4. THE STEADY STATE PROPERTIES OF ATOMS IN THE SQUEEZED VACUUM...... 50

4.1 Quantum entanglement of two qubits ...... 50 4.2 Resonance ﬂuorescence of a group of atoms ...... 54 4.3 The steady state population inversion of a single Ξ-type atom by the squeezed vacuum 58 4.4 The steady state population inversion of multiple Ξ-type atoms by the squeezed vacuum ...... 60

5. CAVITY-CAVITY INTERACTION IN THE SQUEEZED VACUUM ...... 69

5.1 General master equation of cavity-cavity interaction...... 69 5.2 Steady state of non-resonant cavities...... 74

viii 5.3 Steady state of resonant cavities ...... 76

6. SUMMARY AND CONCLUSIONS...... 79

REFERENCES ...... 81

ix LIST OF FIGURES

FIGURE Page

2.1 The error ellipse for the squeezed vacuum...... 6 2.2 An SPDC scheme with the Type I output...... 8 2.3 The photon number distribution for the single mode squeezed vacuum...... 9 2.4 An SPDC scheme with the Type II output...... 10 2.5 The photon number distribution for the two-mode squeezed vacuum...... 12 2.6 The phase matching condition in SPDC process...... 15 3.1 Demonstration for squeezing all modes in a single direction. The Electromagnetic wave propagates in two opposite directions, so two pumps are needed to squeeze allmodes...... 23 3.2 (a) Schematic setup for waveguide-QED in the 1D squeezed vacuum where the vacuum is squeezed from both directions. (b) The dispersion relations inside the 1.2cπ waveguide. Here the atomic transition frequency is a , which is below the cutoff frequency of TE11 mode. Considering the fact that the atomic dipole moment is along y-axis and Ey 6= 0 only for TE10, we only need to consider TE10 mode in our calculation...... 32 3.3 (a) The dephasing dynamics of a single emitter in the squeezed vacuum. The black and red solid curves are the results of σx and σy, respectively. The blue dotted line is the result when there is no squeezing (thermal reservoir). (b) The dephasing rates of σx and σy as a function of the emitter position. For (a)&(b), the squeezing parameters are chosen to be r = 0.5...... 39 3.4 Two-emitter case: Transverse polarization decay of the first emitter as a function of time. (a) r12 = 0.5λ0z for superscript (1) and r12 = 1.0λ0z for superscript (2), r = 0.5 and rc = 0; (b) population decay as a function of time when r12 = 0.5λ0z, r = 0.5 and rc = 0. Solid lines are the results in squeezed vacuum and the dotted lines are the results in the thermal reservoir with N = sinh2(r). Here the dynamics of ρ++ and ρ−− are highly identical. (c) Dephasing rate as a function of atom separation with the center of mass fixed at rc = 0. (d) Dephasing rate as a function of center of mass position with atom separation fixed at rij = λ0z, where the two- atom case is plotted in solid lines and the five-atom case is plotted in dashed lines. .. 42 4.1 (a) Concurrence evolution of different initial states in squeezed vacuum, where r = 1, rc = 0, and r12 = 0.25λ0z. (b) Fidelity evolution of different initial states in the same environment...... 52

x 4.2 (a) Concurrence of the steady state as a function of average photon number N = 2 r1+r2 sinh(r) and the position of the center mass rc = 2 .(b) The impact of rc’s ﬂuctuations on concurrence for different average photon number N. ∆rc is the n distance from 4 λ0z to the position where the entanglement vanishes...... 55 4.3 Resonance ﬂuorescence spectrum of the two-emitter system inside a 1D waveguide. For better comparison, the spectra are normalized to the intensity at ω = ω0 with the coherent elastic scattering singularity removed. Coherent driving Rabi frequency is ΩR = 4γ. In (a) and (b), the solid curves are the spectra for the coupled emitters, while the dashed curves are the spectra without emitter-emitter coupling. Parameters: (a) r1 = 0, r2 = 0.01λ0z, squeezing parameter r = 0.5. (b) r1 = 0, r2 = 0.25λ0z, r = 0.5. (c) r1 = 0, r2 = λ0z, φ = π/2, r = 0.5 for black line, r = 1 for red line. (d) r1 = −0.125λ0z, r2 = 0.125λ0z for the red line, r1 = −0.25λ0z, r2 = 0.25λ0z for the black line. φ = 0, r = 0.5...... 56

4.4 (a) The steady state population distribution for different µab and µbc. The squeezing parameter r = 1 and the squeezing is perfect (M = pN(N + 1)). (b) The steady state population distribution for non-ideal squeezed vacuum which is characterized p 1 by the ratio of M and N(N + 1). The squeezing parameter r = 1, and γab = 4 γbc. 61 4.5 The allowed population ﬂow in the squeezed vacuum...... 62 4.6 (a) Fidelity evolution with different atomic separations. The atomic separations of γ1 1 λ0, 0.1λ0, 0.2λ0 are plotted. Squeezing parameter r = 1, decay rate = and √ γ2 4 time unit τ = 1/ γabγbc is the geometric mean of the transition |ai → |bi and |bi → |ci’s spontaneous emission rates in ordinary vacuum. (b) Fidelity evolution with different squeezing parameters. Decay rate γ1 = 1 , and atomic separation γ2 4 r12 = λ0. (c) Fidelity evolution with different decay rates. Squeezing parameter r = 1, and atomic separation r12 = λ0...... 64 5.1 (a) Schematic setup: two single-mode cavities are placed inside the waveguide with the broadband squeezed vacuum incident from both ends...... 74

xi 1. INTRODUCTION1

Due to the well known Purcell effect [1], the spontaneous decay rate of an emitter can be modiﬁed by engineering the electromagnetic bath environment with which the emitters interact. One example of bath engineering is the squeezed vacuum. Although the squeezed vacuum does not change the density of the electromagnetic modes, it can still modify the decay rate of the emitter [2, 3, 4]. A single emitter interacting with the squeezed vacuum has been widely studied [5, 6, 7]. However, there are only a few publications dealing with multiple emitters interacting with squeezed vacuum. Among these works, most are considering the case where emitters are separated by much less than an optical wavelength which is the well known Dicke model [8]. It is shown that in a broadband squeezed vacuum, emitter system evolves into a state whose properties are similar to those of the squeezed vacuum. Only a very few papers study the case when the separation between the emitters becomes important [9, 10, 11]. It is found that the dipole-dipole interaction induced by ordinary vacuum depends on the relative emitter separation, while the interaction induced by the squeezed vacuum depends on the center of mass coordinate of the emitters. Since it depends on the position of the center of mass, the choice of the coordinate system should be no longer arbitrary. However, it is not yet clearly illustrated in these literature on how to choose the coordinate system. Actually, the dependence on the absolute position comes from the fact that the squeezed vacuum is not vacuum but generated by a coherent light source. The phase of a coherent source is important for the dynamics of the emitter system [12] and it is seldom considered in the previous literature[9, 10, 11]. People usually thought this phase can be included in the phase of the correlation function. However, the phase in the correlation function is usually treated as a constant, while it can be a function of position. In addition, the previous calculations mainly consider a broadband squeezing in all directions of the 3-dimensional (3D) space which is difﬁcult to be experimentally realized.

1Parts of the Abstract and this section are reprinted with permission from: “Waveguide QED in the Squeezed Vacuum” by Jieyu You et al, 2018. Physical Review A, 97, 023810, Copyright 2018 by the American Physical Society and “Steady-state population inversion of multiple Ξ-type atoms by the squeezed vacuum in a waveguide” by Jieyu You, Zeyang Liao, and M. Suhail Zubairy, 2019. Physical Review A, 100, 013843, Copyright 2019 by the American Physical Society

1 Recently, photon transport in a one-dimensional (1D) waveguide coupled to quantum emitters (well known as “waveguide-QED") has attracted much attention due to its possible applications in quantum device and quantum information [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In these previous studies, the photon modes in the waveguide are usually considered to be ordinary vacuum modes. The case when the waveguide modes are squeezed is seldom studied. In contrast to the 3D case, squeezing in 1D is more experimentally feasible. Suppression of the radiative decay of atomic coherence and the linewidth of the resonance fluorescence have been experimentally demonstrated in a 1D microwave transmission line coupled to single artificial atom [26, 27, 28, 29]. However, many-body interaction in a 1D waveguide-QED system coupled to squeezed vacuum has not yet been studied. In this dissertation we consider the phase of the squeezing source and rederive the master equation for multi-atom dynamics in the squeezed vacuum based on the Weisskopf-Wigner approximation. We show that while the collective dipole-dipole interaction due to the ordinary vacuum depends on the emitter separation, the collective two-photon decay rate due to the squeezed vacuum largely depends on the center of mass position of the emitters relative to the squeezing source. We then apply this theory to the 1D waveguide-QED system with squeezing reservoir. Contrary to the traditional result that the dephasing rate of a single atom in the squeezed vacuum is a constant [4, 30], our calculation shows that the dephasing rate is actually position-dependent. As dipole-dipole interaction is involved, both emitter separation and center of mass coordinate can affect the decay rate, dephasing rate and the emitted resonance fluorescence spectrum. In addition, we also show that stationary quantum entanglement can be prepared in this system by the squeezing reservoir. The stationary maximum entangled NOON state can be approached if the center-of-mass of the emitters is at certain position. Considering the fact that two qubits can be treated as a four-level system, the stationary NOON state implies the occurrence of population inversion in the steady state. The concept of population inversion is of fundamental importance in laser physics because the population inversion is a key step of generating laser. However, the population inversion can never exist for a system at thermal

2 equilibrium because of the spontaneous emission. The achievement of population inversion therefore requires pushing the system into a non-equilibrated state [31]. Thus, the spontaneous emission must be inhibited in order to maintain the population inversion in a steady state. In 1946, Purcell showed that the spontaneous decay rate of an emitter can be modified by engineering the electromagnetic bath environment with which the emitters interact [1]. One famous example of bath engineering is the squeezed vacuum which leads to many novel effects and techniques in quantum optics and atomic spectroscopy. The reduction of quantum fluctuations below vacuum level by the squeezed vacuum yields many interesting phenomenons, for example, the suppression of dephasing rate in one direction and enhancment in the other for a two-level emitter [2, 3, 4, 7, 8, 9, 10, 11], the subnatural linewidth of resonance fluorescence [32, 29], and improvement of an atomic clock using squeezed vacuum [33]. The entanglement nature of the squeezed vacuum also leads to interesting results like pairwise excitation of atomic states [34, 35, 36]. In 1993, Ficek and Drummond studied the dynamical properties of a single three-level atom in the squeezed vacuum where they showed that a single three-level atom in the cascade configuration coupled to squeezed modes in a cavity can reach steady state with level population inversion relative to the ordinary laser spectroscopy [37, 38, 39]. In their model, they found a population inversion of about 78%. In this dissertation, we consider multiple Ξ-type atoms coupled to a broadband squeezed vacuum in the quasi-1D waveguide where all resonant modes can be technically squeezed. We show that for a single atom, it can always reach a population inversion of almost 100% or any other ratio as long as the direction of its transition dipole moment is properly set. We also mathematically prove that this result can be generalized to arbitrary number of atoms coupled to each other through dipole-dipole interaction, which may be a scenario for studying two-photon laser or collective atomic effect.

3 2. SQUEEZED VACUUM RESERVOIR AND TRADITIONAL RESERVOIR THEORY

In this section, we will ﬁrst introduce the basic properties of the squeezed vacuum and why it draws so much interests from researchers. Then we will discuss the traditional way to study the interaction between the atomic system and the squeezed vacuum reservoir. Finally, we will generalize the theory to multi-atom system where the dipole-dipole interaction is included.

2.1 Introduction to the squeezed vacuum

If two operators satisfy the commutation relation [A,ˆ Bˆ] = iCˆ, then according to the Heisen- berg uncertainty relation, it follows that

D ED E 1 (∆Aˆ)2 (∆Bˆ)2 ≥ |hCˆi|2 (2.1) 4

D E D E ˆ 2 1 ˆ ˆ 2 1 ˆ Thus, either (∆A) ≥ 2 |hCi| or (∆B) ≥ 2 |hCi| must be satisfied. A state is defined to be squeezed if one of the following is satisfied:

D E 1 D E 1 (∆Aˆ)2 < |hCˆi| or (∆Bˆ)2 < |hCˆi| (2.2) 2 2

Consider the case for quadrature operator where

ˆ ˆ 1 † A = X1 = aˆ +a ˆ 2 (2.3) 1 Bˆ = Xˆ = aˆ − aˆ† 2 2i

then we have h i i Xˆ , Xˆ = (2.4) 1 2 2

2 2 ˆ ˆ For a coherent state |αi, we have ∆X1 = ∆X2 = 1/4 which is the same as that in ordinary vacuum. Thus, quadrature squeezing, which satisﬁes one of the equations in Eq.(2.2), has even less noise than vacuum. One way to generate such a quadrature squeezing state is hitting

4 the squeeze operator 1 Sˆ(ξ) = exp ξ∗a2 − ξa†2 (2.5) 2 on vacuum state |0i, which is called "the single mode squeezed vacuum". The ﬂuctuation of the squeezed vacuum can be calculated as follows:

2 1 ∆Xˆ = cosh2 r + sinh2 r − 2 sinh r cosh r cos θ 1 4 (2.6) 2 1 ∆Xˆ = cosh2 r + sinh2 r + 2 sinh r cosh r cos θ 2 4

ˆ ˆ To study the role of θ, we deﬁne the rotated quadrature operators Y1 and Y2 as follows:

      ˆ ˆ Y1 cos θ/2 sin θ/2 X1   =     (2.7)  ˆ     ˆ  Y2 − sin θ/2 cos θ/2 X2

Substituting Eq.(2.6) , we have 2 1 ∆Yˆ = e−2r 1 4 (2.8) 2 1 ∆Yˆ = e2r 2 4

which can be represented in Fig. 2.1. Thus, for the squeezed vacuum, squeezing occurs along θ/2 in the phase space. The most famous approach to generate the above squeezed vacuum is based on degenerate parametric process through nonlinear optical crystal, which is shown in Fig. 2.2 . A degenerate

parametric down-converter pumped by a ﬁeld of frequency ωP can split photons of that ﬁeld into a

pair of entangled "signal" photons with the same frequency ωP /2. This process is called degenerate parametric down-conversion with the Hamiltonian as below:

ˆ † ˆ†ˆ (2) 2ˆ† †2ˆ H = ~ωaˆ aˆ + ~ωPb b + i~χ aˆ b − aˆ b (2.9)

where b is the pump mode operator, a is the signal mode operator, and χ(2) is the second-order

5 Figure 2.1: The error ellipse for the squeezed vacuum. (a) θ = 0. (b) A general value of θ. Reproduced with permission of The Licensor through PLSclear. Original author: Christopher Gerry, Peter Knight, Introductory Quantum Optics. Copyright by Cambridge University Press.[40]

6 nonlinear susceptibility of the device. When the pump ﬁeld is a strong classical ﬁeld, we can

make the "parametric approximation" whereby the operators ˆb and ˆb† can be replace by βe−iωPt and β∗eiωPt, respectively. Thus, the Hamiltonian in Eq.(2.9) can be reduced to

(PA) † ∗ 2 iω t †2 −iω t Hˆ = ~ωaˆ aˆ + i~ η aˆ e P − ηaˆ e P (2.10)

with η = χ(2)β. This Hamiltonian can be simpliﬁed further in the interaction picture since energy

is conserved ωP = 2ω: ˆ ∗ 2 †2 HI = i~ η aˆ − ηaˆ (2.11) ˆ ˆ ∗ 2 †2 Thus, the associated evolution operator U1(t) = exp −iH1t/~ = exp η taˆ − ηtaˆ is exactly the squeezed operator in Eq.(2.5) with ξ = 2ηt. The degenerate four-wave mixing where two pump photons are converted into two signal photons of the same frequency can also be used to generate the squeezed vacuum, which relies on the third order nonlinear susceptibility of the device. The calculation is almost identical to the above process. The photon number statistics of the above squeezed vacuum is interesting. Hitting Eq. (2.5) on vacuum gives

∞ p 1 X (2m)! |ξi = √ (−1)m eimθ(tanh r)m|2mi (2.12) 2mm! cosh r m=0

whose photon number probability vanishes for all odd photon numbers, as shown in Fig. 2.3. Quantum damping is a very significant topic in quantum optics, which is caused by interacting with a system with a large number of degrees of freedom. Such a system is called reservoir. For example, Atomic decay and decoherence is caused by electromagnetic interaction between atomic dipole and radiative fields. Engineering the reservoir requires modifying the infinite number of modes in the reservoir. For vacuum, all the electromagnetic modes are in the ground state with zero photon. For black body radiation in a thermal equilibrium at temperature T , the photon number distribution in each mode can be described by the Bose-Einstein distribution[30]. For the single- mode squeezed vacuum, it is implausible to engineer a reservoir with all modes in the single-mode squeezed vacuum state because different modes requires different materials and pump fields of

7 Figure 2.2: An SPDC scheme with the Type I output. This ﬁgure is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license: https: //en.wikipedia.org/wiki/File:Scheme_of_spontaneous_parametric_ down-conversion.pdf[41].

8 Figure 2.3: The photon number distribution for the single mode squeezed vacuum. Reproduced with permission of The Licensor through PLSclear. Original author: Christopher Gerry, Peter Knight, Introductory Quantum Optics. Copyright by Cambridge University Press.[40]

9 Figure 2.4: An SPDC scheme with the Type II output. This ﬁgure is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license: https://en.wikipedia.org/ wiki/File:Scheme_of_spontaneous_parametric_down-conversion.pdf[41]

10 different frequency. To engineer such a squeezed vacuum reservoir, we need two-mode squeezed vacuum. The two-mode squeezed vacuum can be generated by the spontaneous parametric down- conversion process described by the following Hamiltonian:

ˆ † ˆ†ˆ † (2) ˆ † †ˆ† H = ~ωaaˆ aˆ + ~ωbb b + ~ωPcˆ cˆ + i~χ aˆbcˆ − aˆ b cˆ (2.13) which is the non-degenerate form of (2.10), with the process shown in Fig. 2.4. We can impose the same parametric approximation to replace c by γe−iωP t and deﬁne η = χ(2)γ. Then the Hamilto- nian becomes ˆ (PA) † ˆ†ˆ ∗ iωPt ˆ −iωPt †ˆ† H = ~ωaaˆ aˆ + ~ωbb b + i~ η e aˆb − ηe aˆ b (2.14)

Considering the energy should be conserved for this three-wave mixing process, we have the following Hamiltonian in the interaction picture:

ˆ ∗ ˆ †ˆ† HI = i~ η aˆb − ηaˆ b (2.15)

Thus, the associated evolution operator is

ˆ h ˆ i U1(t, 0) = exp −iH1t/~ (2.16)

Deﬁning ξ = ηt, we have the two-mode squeezed vacuum

ˆ ˆ ∗ ˆ †ˆ† |ξi2 = U1(t, 0)|ξi2 = S2(ξ)|0, 0i = exp ξ aˆb − ξaˆ b |0, 0i (2.17)

Expanding the exponential operator, we have

∞ 1 X |ξi = (−1)neinθ(tanh r)n|n, ni (2.18) 2 cosh r n=0

It is interesting that this two-mode squeezed vacuum state is a photon number entangled state with photons in the correlated modes always generated by pair, as shown in Fig. 2.5

11 Figure 2.5: The photon number distribution for the two-mode squeezed vacuum. Reproduced with permission of The Licensor through PLSclear. Original author: Christopher Gerry, Peter Knight, Introductory Quantum Optics. Copyright by Cambridge University Press.[40]

12 However, when we only focus on one mode by taking the partial trace of the other mode, we can ﬁnd that each mode seems to be in the thermal state:

∞ X ρˆa = hnb|ξi2hξ|2|nbi

nb=0 ∞ (2.19) 1 X = (−1)na einaθ(tanh r)na |n ihn | cosh r a a na=0 whose photon number distribution is

(tanh r)2n P = hn |ρˆ| ni = (2.20) n (cosh r)2

which is exactly the Boltzmann distribution with the average photon number hni = sinh2 r. The two-mode squeezed vacuum is also a squeezed state. Considering the following quadrature operators: 1 Xˆ = aˆ +a ˆ† + ˆb + ˆb† 1 23/2 (2.21) ˆ 1 † ˆ ˆ† X2 = aˆ − aˆ + b − b 23/2î h ˆ ˆ i which satisfies the commutation relation X1, X2 = i/2, we have the fluctuations as follows:

2 1 Xˆ = cosh2 r + sinh2 r − 2 sin r cosh r cos θ 1 4 (2.22) 2 1 Xˆ = cosh2 r + sinh2 r + 2 sin r cosh r cos θ 2 4

This is identical to Eq. (2.6). Thus, the two-mode squeezed vacuum is also a squeezed state. If all modes are squeezed in the above way, the reservoir is called the squeezed vacuum reservoir. However, it is worth nothing that generating the two-mode squeezed vacuum in all modes with

the same mid frequency ω0 and the same squeezing parameter r is still experimentally impossible due to many reasons. The most signiﬁcant reason is that, both the three-wave mixing and four- wave mixing must satisfy the phase-matching condition so that both energy and momentum must be conserved, as depicted in Fig. 2.6. Thus, it is theoretically impossible to squeeze all modes in all directions in the 3D space with one pump ﬁeld. Furthermore, even those squeezed modes

13 do not have a uniform squeezing parameter r, since the second-order nonlinear susceptibility χ(2) depends on the frequency of the output ﬁeld. Hence in experiments, the concept of "broadband squeezed light" is used. Instead of the single-mode quadrature operators in Eq. (2.3) or two-mode quadrature operators in Eq. (2.21), collective quadrature operators are considered:

h i ˆ (c) 1 ˆ(+) ˆ(−) X1 (t) = E (t) + E (t) 2 (2.23) 1 h i Xˆ (c)(t) = Eˆ(+)(t) − Eˆ(−)(t) 2 2i

q where the annihilation part is Eˆ(+)(r, t) = i P ~ω e(s)aˆ(s)(k)eik·r−iωkt and the creation part k,s 2V 0 q is Eˆ(−)(r, t) = −i P ~ω e(s)aˆ†(s)(k)e−ik·r+iωkt. If we have the following commutation rela- k,s 2V 0 tion measured from experiments

h i Eˆ(+)(t), Eˆ(−)(t) = C (2.24)

then we have h i i Xˆ (c), Xˆ (c) = C (2.25) 1 2 2

Thus, the broadband squeezing is deﬁned if

2 1 ∆Xˆ (c) < |C| i 4

is satisﬁed for i = 1 or i = 2.

2.2 Traditional reservoir theory

Considering the fact that a reservoir is a system with a large number of degrees of freedom, the reservoir can be interpreted as an open system. Thus, if an atom in the excited state interacts with the reservoir, the atom will decay to the ground state with photon emitted to the reservoir and never re-absorbed by the atom, which is called quantum damping. Since the reservoir requires so many degrees of freedom to be fully described, we can apply Markovian assumption on this process that

14 Figure 2.6: The phase matching condition in SPDC process. This ﬁgure is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license: https://upload.wikimedia.org/wikipedia/commons/7/74/Spontaneous_ Parametric_Downconversion.png[41].

the damping destroys memory of the past. Thus, for a system represented as ρS interacting with a reservoir represented as ρF , its evolution can be described by the following equation[30]:

i ˙S S F ρ = − T rR[V (t), ρ (0) ⊗ ρ (0)] ~ t Z (2.26) 1 S F − T rR [V (t), [V (t − τ), ρ (t − τ) ⊗ ρ (0)]]dτ ~2 0

where V (t) is the interaction Hamiltonian in the interaction picture. For a two-level system (qubit) in free space, the interaction Hamiltonian is

† ∗ − ∗ † V (t) = d · E = ~[µS (t) + µ S (t)] · [uks(ri)ˆaks(t) + uks(ri)ˆaks(t)] (2.27)

15 where S− = |biha| is the lowering operator and S+ = |aihb| is the raising operator of the qubit.

± ± ±iω0t −iωkst † † iωkst In the interaction picture, we have Si (t) = Si e , aˆks(t) =a ˆkse , and aˆks(t) =a ˆkse . uks(ri) is the mode function which is typically given by

r ω k,s ik·ri uk,s(ri) = ek,se (2.28) 20~V where s indicates the polarization. For a thermal reservoir which is described by the density operator † ! Y ~vk ~vkbkbk ρR = 1 − exp − exp − (2.29) kBT kBT k,s

where kB is the Boltzmann constant and T is the thermal equilibrium temperature, we have

D † E hak,si = ak,s = 0

D † E ak,sak0,s0 =n ¯k,sδk0kδss0 (2.30) D † E ak,sak0,s0 = (¯nk,s + 1)δk0kδss0

D † † E ak,sak0,s0 = hak,sak0,s0 i = 0

1 where n¯k,s is the average photon number with n¯k,s = v . Substituting them into Eq.(2.26) exp ~ k −1 kBT and applying Wigner-Weisskopf approximation, we have

dρS 1 = − γ(N + 1) ρSS+S− + S+S−ρS − 2S−ρSS+ dt 2 (2.31) 1 − γN ρSS−S+ + S−S+ρS − 2S+ρSS− 2

16 1 4ω3µ2 where γ = 3 is the spontaneous decay rate and N is the average photon number in the 4π0 3~c resonant frequency mode. Analyzing each element in Eq.(2.31), we have

ρ˙aa = − (N + 1) γρaa + Nγρbb 1 ρ˙ =ρ ˙∗ = − N + γρ (2.32) ab ba 2 ab

ρ˙bb = −Nγρbb + (N + 1) γρaa

At T = 0, this reduces to

ρ˙aa = −γρaa

γ (2.33) ρ˙ab = − 2 ρab

ρ˙bb = γρaa

In general, for Na multi-level atoms interacting with the electromagnetic ﬁeld, the total Hamil- tonian can be written as

H = HA + HF + HAF (2.34)

where the atomic free Hamiltonian is

N Xa X HA = ~ωe,l |eli hel| (2.35) l=1 e=a,b,c

with |eli representing the energy state of the lth atom with energy ~ωe,l. The Hamiltonian of the EM ﬁeld is X 1 H = ω (ˆa† aˆ + ) (2.36) F ~ ks ks ks 2 ks

† where aˆks and aˆks are the annihilation and creation operators of the ﬁled mode with wavevector k and polarization s. The interaction Hamiltonian in the electric-dipole approximation is

Na X X X + ∗ − HAF = −i~ [µl,i · uks(rl,i)Sl,iaˆks + µl,i · uks(rl,i)Sl,iaˆks − H.c.] (2.37) ks i l=1

where i denotes the ith atomic transition. Transforming the total Hamiltonian H into the interac-

17 tion picture, and substituting them into Eq.(2.26), considering the thermal reservoir described in Eq.(2.30), we have the following equation[42]: (The calculation will be derived in a very detailed manner in section 2, here we just jump to the conclusion)

dρS X = − i Λ [S+ S− , ρS]ei(ωj −ωl)t dt ijkl i,j k.l ijkl 1X − γ (1 + N)(ρSS+ S− + S+ S− ρS − 2S− ρSS+ )ei(ωj −ωl)t 2 ijkl i,j k.l i,j k.l k.l i,j (2.38) ijkl 1X − γ N(ρSS− S+ + S− S+ ρS − 2S+ ρSS− )e−i(ωj −ωl)t 2 ijkl i,j k.l i,j k,l k,l i,j ijkl

The collective energy shifts Λijkl and decay rates γijkl due to the ordinary vacuum are given by [43, 42] 3√ 2 cos(k0rik) Λijkl = γijγkl{−(1 − cos α) 4 k0rik sin(k r ) cos(k r ) 2 0 ik 0 ik (2.39) + (1 − 3 cos α)[ 2 + 3 ]} (k0rik) (k0rik) √ γijkl = γijγklF (k0rik)

where subscripts i, k label the atom index, j(l) labels the transitions of the ith(kth) atom, γ =

3 2 ω0 µ 3 2 sin x 3 is the spontaneous decay rate of the atom in ordinary vacuum and F (x) = {(1−cos α) + 3π0~c 2 x 2 cos x sin x (1 − 3 cos α)[ x2 − x3 ]}. However, if we apply the above calculations to a single atom interacting with the squeezed vacuum, we will have some weird result. Since we have

D † E hak,si = ak,s = 0

D † E 2 ak,sak0,s0 = sinh rδk0kδss0

D † E 2 0 0 (2.40) ak,sak0,s0 = cosh rδk kδss

D † † E −iθ 0 0 ak,sak0,s0 = −e cosh(r) sinh(r)δk ,2k0−kδss

iθ 0 0 0 0 hak,sak ,s i = −e cosh(r) sinh(r)δk ,2k0−kδss

18 Substituting them into Eq.(2.26), we have

dρS = sinh(r) cosh(r)γ0(S+ρSS+ + H.c.) dt 1 − γ cosh2(r)(ρSS+S− + S+S−ρS − 2S−ρSS+) (2.41) 2 1 − γ sinh2(r)(ρSS−S+ + S−S+ρS − 2S+ρSS−) 2

0 with γij = γF (2k0r)). However, although the position of atom is well-deﬁned, its coordinate is ill-deﬁned since there is no requirement on the origin of the coordinate system. This value varies from 0 to γ by the value of r, which is not physical. For example, if we choose r = 0, then

0 γij = γ which is the perfect squeezing; if r is the zero point of F (2k0r), Eq.(2.41) reduces to the thermal reservoir as Eq.(2.31). As we mentioned before, the choice of r completely depends on the coordinate system, which can be built arbitrarily. Thus, the above theory cannot be applied on the squeezed vacuum, and putting forward a modiﬁed theory is the main goal of this dissertation.

19 3. A MODIFIED RESERVOIR THEORY IN THE SQUEEZED VACUUM1

In the last section, we discussed about the limitation of the traditional reservoir theory when it is applied on the squeezed vacuum. In this section, we will bring forward a modiﬁed theory. We will show that this theory works properly with the squeezed vacuum, and remains consistent with the traditional theory in the thermal reservoir case. We will discuss the new behaviors of the system based on our new theory.

3.1 A new master equation from a modiﬁed mode function

First of all, we need to analyze the origin of the unphysical quantity γ0 in Eq.(2.41), and here we start from the most typical single-qubit case with transition frequency ω. The interaction Hamilto- nian is

X + ∗ − V (t) = −i~ [µ · uks(r)S (t)ˆaks(t) + µ · uks(r)S (t)ˆaks(t) − H.c.] (3.1) ks with the mode function r ωks ik·r uks(r) = ekse (3.2) 20~V

D † E Considering that hak,si = ak,s = 0 for both thermal reservoir and the squeezed vacuum reservoir, we can drop the ﬁrst term in Eq. (2.26), so we have the master equation

S Z t dρ 1 S F = − 2 dτT rF {[V (t), [V (t − τ), ρ (t − τ)ρ } dt ~ 0 Z t 1 S F S F (3.3) = − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ + ρ (t − τ)ρ V (t − τ)V (t) ~ 0 − V (t)ρS(t − τ)ρF V (t − τ) − V (t − τ)ρS(t − τ)ρF V (t)}.

1Part of this section is reprinted with permission from: “Waveguide QED in the Squeezed Vacuum” by Jieyu You et al, 2018. Physical Review A, 97, 023810, Copyright 2018 by the American Physical Society and “Steady-state population inversion of multiple Ks-type atoms by the squeezed vacuum in a waveguide” by Jieyu You, Zeyang Liao, and M. Suhail Zubairy, 2019. Physical Review A, 100, 013843, Copyright 2019 by the American Physical Society.

20 which is of the second order in the interaction Hamiltonian V (t) and mode function uks(ri). To make the derivation simple in form, we deﬁne

† ∗ − D(t) = [µ · uk,s(r)S (t) + µ · uk,s(r)S (t)] (3.4) so the interaction Hamiltonian Eq.(3.1) becomes

X + † V (t) = −i~ [D(t)aks(t) − D (t)aks(t)] (3.5) ks

Here we just show the way to deal with the ﬁrst term in Eq.(3.3), the remaining terms can be calculated in the same way. For the ﬁrst term, we have

Z t 1 S F − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ } ~ 0 Z t X F + F † = dτ {D(t)D(t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] − D(t)D (t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] 0 ks,k0s0

+ F † + + F † † S − D (t)D(t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] + D (t)D (t − τ)T rF [ρ aks(t)ak0s0 (t − τ)]}ρ (t − τ)}. (3.6)

21 Using the correlation functions Eq.(2.40), we have the following result under the rotating wave approximation (RWA):

Z t 1 S F − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ } ~ 0 Z t X + iωt + iω(t−τ) −i(ωks+ω 0 0 )t+iω 0 0 τ = dτ{µ · uks(r)S e µ · uk0s0 (r)S e e k s k s ks,k0s0 0

0 0 × [− sinh(r) cosh(r)δk ,2k0−kδss ]

+ iωt ∗ ∗ − −iω(t−τ) −iωk0s0 τ 2 − µ · uks(r)S e µ · uk0s0 (r)S e e cosh rδkk0 δss0 (3.7) ∗ − −iωt ∗ + iω(t−τ) −iωk0s0 τ 2 − µ · uks(r)S e µ · uk0s0 (r)S e e cosh rδkk0 δss0

∗ ∗ − −iωt + iω(t−τ) iωk0s0 τ 2 − µ · uks(r)S e µ · uk0s0 (r)S e e sinh rδkk0 δss0

∗ + iωt ∗ − −iω(t−τ) iωk0s0 τ 2 − µ · uks(r)S e µ · uk0s0 (r)S e e sinh rδkk0 δss0

∗ ∗ − −iωt ∗ ∗ − −iω(t−τ) i(ωks+ωk0s0 )t−iωk0s0 τ + µ · uks(r)S e µ · uk0s0 (r)S e e

S 0 0 × [− sinh(r) cosh(r)δk ,2k0−kδss ]}ρ (t − τ)

∗ For the second to the ﬁfth terms, uks(r)uk0s0 (r)’s positional dependence is canceled. However, the ﬁrst depends on eik·r and the sixth term depends on eik·r. This is not an issue for the thermal

† † reservoir since hak,sak0,s0 i = hak,sak0,s0 i = 0 so these two terms vanish. For the squeezed vacuum, these terms are positional dependent. What’s more, since there is no constraint on the choice of coordinate system, r can be arbitrary which yields the arbitrary value of the first and sixth terms in the above equation. This unphysical behavior originates from the definition of the mode function in Eq.(3.2). However, this formula is so famous that it appears in almost every textbook on quantum mechanics and quantum mechanics, so challenging it should be very cautious. We noticed that the usage of Eq. (3.2) is always related to the quantization of radiation field with the running wave boundary condition, for example in Quantum Optics by M.O. Scully and M.S. Zubairy[30]. However, if the standing wave boundary condition is applied, eik·r should be replace by cos(k·r) or sin(k·r). Thus, equation (3.2) can only be used when the space translational symmetry is satisfied. In the thermal reservoir, the space translational symmetry is satisfied. However, this is not the case

22 Figure 3.1: Demonstration for squeezing all modes in a single direction. The Electromagnetic wave propagates in two opposite directions, so two pumps are needed to squeeze all modes.

for the squeezed vacuum reservoir. Considering the squeezed vacuum reservoir in one dimensional case, two pump beams interacting with the nonlinear crystal from opposite directions are needed to form a broad-band squeezed vacuum, which is shown in Fig. 3.1. Due to the existence of the nonlinear crystal and the pump ﬁeld, the space translational symmetry is broken. Thus, we cannot use Eq. (3.2) as the mode function. Here we propose a new mode function which incorporate both the existence of the squeezing source and the nature of running wave of the squeezed vacuum:

r ω k,s ik·(ri−ok,s) uk,s(ri) = ekse (3.8) 20~V

where oks includes the effects of the initial phase and the position of the squeezing source with wavevector ks. Here we need to make two assumptions: first, one specific mode is generated from a single source, i.e., mode ks is only generated from the source located at oks; second, the phases of all modes can be well defined by k · (r − oks). In the ordinary vacuum or thermal reservoir, there is no source and we can set oks = 0, so the mode function shown in Eq. (3.8) is reduced to the normal cases as Eq. (3.2). In fact, oks can be any value since their effects will be canceled in Eq. (3.6). Considering the fact that the squeezed vacuum is generated by the pump field from different directions, oks should be different for different modes. Next, we will derive the master equation from Eq. (3.6) based on the new mode function

23 Eq.(3.8). We start from a more general case where atoms are not identical but ωi ≈ ωj, and we P make the squeezing center frequency ω0 = ωi/l. Then we can rewrite the interaction Hamilto- i nian as

X + † V (t) = −i~ [D(t)aks(t) − D (t)aks(t)], (3.9) ks where X † ∗ − D(t) = [µi · uk,s(ri)Si (t) + µi · uk,s(ri)Si (t)] (3.10) i

Therefore, we have

S Z t dρ 1 S F = − 2 dτT rF {[V (t), [V (t − τ), ρ (t − τ)ρ } dt ~ 0 Z t 1 S F S F (3.11) = − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ + ρ (t − τ)ρ V (t − τ)V (t) ~ 0 − V (t)ρS(t − τ)ρF V (t − τ) − V (t − τ)ρS(t − τ)ρF V (t)}.

Here we just show how to deal with the ﬁrst term in Eq.(3.11), the remaining terms can be calculated in the same way. For the ﬁrst term, we have

Z t 1 S F − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ } ~ 0 Z t X F + F † = dτ {D(t)D(t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] − D(t)D (t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] 0 ks,k0s0

+ F † + + F † † S − D (t)D(t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] + D (t)D (t − τ)T rF [ρ aks(t)ak0s0 (t − τ)]}ρ (t − τ)}. (3.12)

Considering that the time scale t0 we care satisﬁes ω0t0 1, the average effects of the oscilla- tions terms like eiωt are ceased. Thus, we impose the rotating wave approximation(RWA), which

24 discards all terms containing eiωt. Using Eq.(3.10) and the correlation function Eq.(2.40), we have

Z t 1 S F − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ } ~ 0 Z t X X + iωit + iωj (t−τ) −i(ωks+ωk0s0 )t+iωk0s0 τ = dτ{µi · uks(ri)Si e µj · uk0s0 (rj)Sj e e ij ks,k0s0 0

0 0 × [− sinh(r) cosh(r)δk ,2k0−kδss ]

+ iωit ∗ ∗ − −iωj (t−τ) −iωk0s0 τ 2 − µi · uks(ri)Si e µj · uk0s0 (rj)Sj e e cosh rδkk0 δss0 (3.13) ∗ − −iωit ∗ + iωj (t−τ) −iωk0s0 τ 2 − µi · uks(ri)Si e µj · uk0s0 (rj)Sj e e cosh rδkk0 δss0

∗ ∗ − −iωit + iωj (t−τ) iωk0s0 τ 2 − µi · uks(ri)Si e µj · uk0s0 (rj)Sj e e sinh rδkk0 δss0

∗ + iωit ∗ − −iωj (t−τ) iωk0s0 τ 2 − µi · uks(ri)Si e µj · uk0s0 (rj)Sj e e sinh rδkk0 δss0

∗ ∗ − −iωit ∗ ∗ − −iωj (t−τ) i(ωks+ωk0s0 )t−iωk0s0 τ + µi · uks(ri)Si e µj · uk0s0 (rj)Sj e e

S 0 0 × [− sinh(r) cosh(r)δk ,2k0−kδss ]}ρ (t − τ)

Assuming all oks are located on the spherical surface with distance D to the origin (0, 0, 0), for the thermal terms (the second to the ﬁfth terms), we can replace the summation of discrete modes with the integral of continuous modes in the free space in the limit of L → ∞:

X L3 Z Z → k2dk (3.14) (2π)3 k Ωk

In Ref. [42], it has been shown that the integral can be calculated as

3 Z Z √ Z ∞ L 2 X ∗ ∗ γiγj 3 k dk µi · uks(ri)µ · u (rj) ≈ dωω F (krij) (3.15) (2π)3 j ks 2πω3 Ωk s 0 0 with

3 2 sin(krij) 2 cos(krij) sin(krij) F (krij) = {[1 − cos α] + [1 − 3 cos α][ 2 − 3 ]} 2 krij (krij) (krij) 3 2 (3.16) ωi µi γi = 3 3π0~c

25 where rij = ri − rj, rij = |rij|, α is the angle between rij and µi, and the approximation in

Eq.(3.15) becomes equality when ω1 = ω2. We can also show that

3 Z Z L 2 X 3 k dk µi · uks(ri)µj · u2k0−k,s(rj) (2π) Ω k s (3.17) √ ∞ γiγj Z k 2p 2ik0R ≈ 3 dωω ω(2ω0 − ω)F (k0| rij + 2rj|)e 2πω0 0 k0

where R is the distance from the sources to the center mass of two atoms, and the approximation

becomes equality when ω1 = ω2. Next, we will show how to calculate the ﬁrst and the second terms in Eq.(3.13), and the remaining terms can be approached in the same way. Using Eq.(3.15), the second term in Eq.(3.13) can be simpliﬁed as

Z t X + iωit ∗ ∗ − −iωj (t−τ) −iωksτ 2 S dτµi · uks(ri)Si e µj · uks(rj)Sj e e cosh rρ (t − τ) 0 ks (3.18) √ t ∞ γiγj Z Z 2 3 i(ωi−ωj )t i(ωj −ωk)τ + − S = cosh r 3 dτ dωω F (krij)e e Si Sj ρ (t − τ) 2πω0 0 0 with F (krij) given in Eq.(3.16). We here calculate the integral of the ﬁrst term in F (krij) (i 6= j) and the other terms can be calculated similarly.

√ 4 t ∞ γiγjc 3 Z Z sinkr 2 3 ij i(ωj −ωk)τ + − S i(ωi−ωj )t cosh r 3 dτ dkk e Si Sj ρ (t − τ)e 2πω0 2 0 0 krij √ 4 t ∞ γiγjc 3 Z Z 1 2 2 i(k−kj )rij +ikj rij −i(k−kj )rij −ikj rij = cosh r 3 dτ dkk (e − e ) 2πω0 2 0 −∞ 2irij

−i(k−kj )cτ + − S i(ωi−ωj )t × e Si Sj ρ (t − τ)e √ 4 t γiγjc 3 Z 1 2 2 ikj rij −ikj rij + − S i(ωi−ωj )t ≈ cosh r 3 dτkj [δ(rij − cτ)e − δ(rij + cτ)e ]Si Sj ρ (t − τ)e 2πω0 2 0 irij √ 4 γiγjc 3 π 2 2 ikj rij + − S i(ωi−ωj )t ≈ cosh r 3 kj e Si Sj ρ (t)e 2πω0 2 icrij 3 eik0rij √ 2 + − S i(ωi−ωj )t ≈ γiγj cosh r Si Sj ρ (t)e 4 ik0rij (3.19)

R ∞ R ∞ In the second line of the equations, we replace 0 dk by −∞ dk since the main contribution comes

26 from the frequency around ω0 and the negative frequency part leads to fast-oscillating term such R t that its integration 0 dτ vanishes. From the second line to the third line, the Weisskopf-Wigner approximation[30] is applied and k is replaced by kj because the contribution comes mainly from the resonant frequency. From the third line to the fourth line, we assume that the two atoms are very close that the time-retarded effect can be neglected. In the last line, we use the fact that

ωi ≈ ω0 The other terms in Eq.(3.18) can be calculated in a similar way, and the result is given by

√ t ∞ γiγj Z Z 1 3 i(ωi−ωj )t i(ωj −ωk)τ + − S + − S i(ωi−ωj )t 3 dτ dkk F (krij)e e Si Sj ρ (t − τ) = ( γij + iΛij)Si Sj ρ (t)e 2πω0 0 0 2 (3.20)

where

3√ cos(k r ) sin(k r ) cos(k r ) Λ = γ γ {−(1 − cos2 α) 0 ij + (1 − 3 cos2 α)[ 0 ij + 0 ij ]} ij 4 i j k r (k r )2 (k r )3 0 ij 0 ij 0 ij (3.21) √ γij = γiγjF (k0rij)

+ − All the other terms with the combination of Si and Si can also be calculated in the same way. Thus, all the thermal terms and oscillation terms in Eq.(3.6) can be given.

+ + − − Next we need to calculate the squeezed vacuum terms containing Si Sj or Si Sj . Here we calculate the ﬁrst term in Eq.(3.6) as an example. Inserting Eq.(3.17), the ﬁrst term of Eq.(3.6) yields

Z t Z X 3 i(ωks−ωj )τ + + S dτ d k{µi · u2k0−k,s(ri)µj · uks(rj)e Si Sj ρ (t − τ) ks,k0s0 0

√ 4 t 2k0 γiγjc Z Z k 2p i(ωk−ωj )τ + + S 2ik0R = 3 dτ dkk k(2k0 − k)F (k0| rij + 2rj|)e Si Sj ρ (t − τ)e 2πω0 0 0 k0 √ t ∞ γiγjc Z Z k i(ωk−ω0)τ + + S 2ik0R ≈ dτ dkF (k0| rij + 2rj|)e Si Sj ρ (t − τ)e 2π 0 −∞ k0 (3.22)

From the second line to the third line, we make the following approximations: (1) The integral limit

27 2p is extended to ±∞ because the principal part is near ωk ≈ ωi. (2) k k(2k0 − k) is pulled out of

3 the integral as a constant k0 according to the Weisskopf-Wigner approximation. To calculate one

term with ﬁxed i, j, we need to rebuild the coordinate system where ri + rj = 0 for i 6= j(We need to build different coordinate systems for different pairs of i, j). For example, we here consider the

1 2 + + S ﬁrst two atoms, i, j = 1, 2. When i = j, this term directly gives 2 γ cosh rF (2k0|rj|)Si Si ρ (t).

When i 6= j, since there is a singular point at k = k0, the calculation is a little bit more complicated but can still be calculated. We have the following integrals:

Z ∞ sin krij −ikcτ π dk e = θ1(rij − cτ), −∞ krij rij Z ∞ (3.23) hcos krij sin krij i −ikcτ π(cτ − rij)(cτ + rij) dk 2 − 3 e = 3 θ2(rij − cτ), −∞ (krij) (krij) 2rij

where θ1,2(x) are step functions: θ1,2(x) = 0 when x < 0, θ1,2(x) = 1 when x > 0, and θ1(0) =

k 1/2 and θ2(0) = 0. Since F (k0| rij + 2rj|) = F ((k − k0)r12), we have k0

Z t Z ∞ i(ω0−ωk)τ S dτ dkF ([(k − k0)r12]e ρ (t − τ) 0 −∞ rij Z c 3 2 π 2 π(cτ − rij)(cτ + rij) S (3.24) = dτ [(1 − cos α) + (1 − 3 cos α) 3 ]ρ (t − τ) 0 2 rij 2rij π ≈ ρS(t). c

In Eq.(3.24), the emitter separation is assumed to be small and the Markovian approximation is

0 S S γij + + S applied such that ρ (t − τ) ≈ ρ (t). Hence, Eq.(3.22) gives sinh r cosh r 2 Si Sj ρ (t) with

0 2ik0R γij = e γF (k0|ri + rj|) after transforming the above results to the original coordinate system(Although replacing k by k0 in Eq.(3.22)’s last line yields the same result, it is not always safe to do so since F (x) is an oscillating function). Having dealt with all the squeezed vacuum terms,

28 we can get the ﬁnal master equation in the 3D free space for arbitrary number of qubits:

dρS 1 X X = − γ0 M(ρSSαSα + SαSαρS − 2SαρSSα) dt 2 ij i j i j j i α=± i,j 1X − γ (1 + N)(ρSS+S− + S+S−ρS − 2S−ρSS+)ei(ωi−ωj )t 2 ij i j i j j i i,j (3.25) 1X − γ N(ρSS−S+ + S−S+ρS − 2S+ρSS−)e−i(ωi−ωj )t 2 ij i j i j j i i,j

X + − S i(ωi−ωj )t − i Λij[Si Sj , ρ ]e i6=j where the last three terms agree with the traditional reservoir theory in the thermal reservoir case, and the ﬁrst term is the collective decay due to the squeezed vacuum. We have M =

2 sinh(r)cosh(r) and average photon number N = sinh (r). The collective energy shifts Λij and decay rates γij due to the ordinary vacuum are the same as those given by [43, 42], which are shown as follow: 3√ 2 cos(k0rij) Λij = γiγj{−(1 − cos α) 4 k0rij

2 sin(k0rij) cos(k0rij) + (1 − 3 cos α)[ 2 + 3 ]} (k0rij) (k0rij) √ γij = γiγjF (k0rij)

3 2 ω0 µ where γ = 3 is the spontaneous decay rate of the atom in ordinary vacuum. Different from 3π0~c the thermal reservoir terms, the squeezed vacuum can contribute to the additional collective two- photon decay rate of the system which is given by

0 2ik0R γij = γe F (k0|ri + rj|). (3.26)

Thus, the collective decay due to the squeezed vacuum depends on the position of the center of mass of the emitters instead of their separation. One may think this reult is identical to the privious

work[9, 10] except the phase e2ik0R, but that is not true. No matter how the coordinate system

is built, to reach the neat form of Eq.(3.26), ri must still be interpreted as the displacement from

29 the center of squeezing sources to the ith atom. When their center of mass is at equal distances from all squeezing sources (i.e., ri + rj = 0), the decay induced by the squeezing is the strongest due to the perfectly constructive interference of the two-photon excitation from all directions. It decreases when it deviates from the center due to the destructive interference. One way to verify the validity of our theory is to prove the positive deﬁnition of Eq.(3.25). The above master equation can be transformed to the Lindblad form [44] and the density matrix is positive. The phase factor e2ik0zR can be effectively regarded as an controllable phase of M, which can be incorporated into θ. The master equation can be transformed as

dρS X X 1 = −i [H, ρS] + h (L ρL† − (ρL† L + L† L ρ)) dt nm n m 2 m n m n (3.27) i m,n where

X + − H = ΛijSi Sj i6=j

+ + − − L1 = S1 ,L2 = S2 ,L3 = S3 ,L4 = S4   2 2 0 0 γ11 sinh r γ12 sinh r γ11 sinh r cosh r γ12 sinh r cosh r (3.28)    2 2 0 0   γ12 sinh r γ11 sinh r γ sinh r cosh r γ sinh r cosh r  12 11  h =   γ0 sinh r cosh r γ0 sinh r cosh r γ cosh2 r γ cosh2 r   11 12 11 12    0 0 2 2 γ12 sinh r cosh r γ11 sinh r cosh r γ12 cosh r γ11 cosh r

0 0 0 0 here for simplicity, we have already used the relations: γ12 = γ21, γ12 = γ21, γ11 = γ22, γ11 = γ22.

0 0 The last relation γ11 = γ22 is not always satisﬁed, but without it we cannot diagonalize matrix h analytically. Hence we set ri + rj = 0. Now matrix h can be diagonalized:

  ζ1      ζ2  †   h = u   u (3.29)  ζ   3    ζ4

30 where u is a unitary matrix, and

1 q ζ = [(γ − γ )(1 + 2 sinh2 r) − (γ − γ )2 + 4 sinh2 r cosh2 r(γ0 − γ0 )2] 1 2 11 12 11 12 11 12 q 1 2 2 2 2 0 0 2 ζ2 = [(γ11 − γ12)(1 + 2 sinh r) + (γ11 − γ12) + 4 sinh r cosh r(γ11 − γ12) ] 2 (3.30) 1 q ζ = [(γ + γ )(1 + 2 sinh2 r) − (γ + γ )2 + 4 sinh2 r cosh2 r(γ0 + γ0 )2] 3 2 11 12 11 12 11 12 1 q ζ = [(γ + γ )(1 + 2 sinh2 r) + (γ + γ )2 + 4 sinh2 r cosh2 r(γ0 + γ0 )2] 4 2 11 12 11 12 11 12

0 0 We noticed that since |γ11 − γ12| = |γ11 − γ12| for ri + rj = 0, none of the eigenvalues is negative,

so the density matrix is completely positive for any initial condition. For arbitrary ri, rj, we can only get the positive eigenvalues numerically. It is worth noting that the master equation derived from the traditional reservoir theories[8, 9, 10] does not have the positive definition in the first place. In fact, the coefficients in their equations are modified by hand to enforce the positive definition.

3.2 Master equation in the quasi-one-dimensional waveguide

In practice, it is very difﬁcult to squeeze all photon modes in 3D case. Since squeezing in 1D is experimentally achievable [28, 29], in this section we discuss the dynamics of the waveguide-QED in the squeezed vacuum. Here, we consider a perfect rectangular waveguide with negligible loss out of the waveguide as is shown in Fig. 3.2(a). We assume that the cross section of the waveguide is a square with dimensions a × b. The origin of the coordinate system is chosen to be at the center of the two squeezing sources with the positions of the sources to be (0, 0, ±R). The emitters are

located along the longitudinal centerline of the waveguide at (0, 0, ri) (i = 1, 2, ··· ,Na) with the squeezed vacuum injected from both ends by the parametric process. Compared with the 3D case,

0 the master equation in the 1D case is the same as Eq. (3.25) except that the values of γij, γij, Λij are different. Different from the free-space case, the square waveguide can only support certain photon modes. Generally, the allowed modes in the waveguide are very complex which need to be ex- pressed in terms of the Dyadic Green functions[45]. Here we consider the simplest case, which

31 (a)

(b)

Figure 3.2: (a) Schematic setup for waveguide-QED in the 1D squeezed vacuum where the vacuum is squeezed from both directions. (b) The dispersion relations inside the waveguide. Here the 1.2cπ atomic transition frequency is a , which is below the cut-off frequency of TE11 mode. Consid- ering the fact that the atomic dipole moment is along y-axis and Ey 6= 0 only for TE10, we only need to consider TE10 mode in our calculation.

32 is the perfect reflective rectangular waveguide with cross section a × b. The rectangular waveguide can support both TE and TM electric field modes and they are given as follows(To get a neat expression of field equation, we set the origin of our coordinate system at the corner of the waveguide):

mπx nπy mπx nπy ETM = E sin sin eikzz,HTE = H cos cos eikzz z 0 a b z 0 a b

TM ikz mπ mπx nπy ikzz TE iωkµ nπ mπx nπy ikzz Ex = E0 2 cos sin e ,Ex = H0 2 cos sin e hmn a a b hmn a a b

TM ikz nπ mπx nπy ikzz TE iωkµ mπ mπx nπy ikzz Ey = E0 2 sin cos e ,Ey = −H0 2 sin cos e hmn a a b hmn a a b

TM iωk nπ mπx nπy ikzz TE ikz mπ mπx nπy ikzz Hx = E0 2 sin cos e ,Hx = −H0 2 sin cos e hmn a a b hmn a a b

TM iωk mπ mπx nπy ikzz TE ikz nπ mπx nπy ikzz Hy = −E0 2 cos sin e ,Hy = −H0 2 cos sin e hmn a a b hmn a a b (3.31)

p mπ 2 nπ 2 where hmn = ( a ) + ( b ) , (µ) is the permittivity (permeability), and H0,E0 are arbitrary

p 2 2 p 2 2 constants. For quantized modes, we have E0 = 4~hmn/ µνLS and H0 = 4~hmn/µ νLS[46].

2 2 2 2 2 The dispersion relation inside the waveguide is given by ωk/c = (mπ/a) + (nπ/b) + kz . For simplicity, we here consider the waveguide with square cross section, i.e., a = b and the dispersion

curves of different modes are shown in Fig. 3.2(b). For square waveguide, TEmn(TMmn) and

TEnm(TMnm) modes are degenerate, and TE10 and TE01 have the lowest energy. We assume that the all emitters’ transition frequencies are the same and they are below the

cutoff frequency of TE11 and TM11 modes. Since the rectangular waveguide cannot support the

TM10 and TM01 mode, the emitter can only couple to the TE01 or TE10 modes. Here, without loss of generality we assume that the transition dipole moment of the emitter is in the y direction.

Thus, it can only couple to the TE10 mode. The emitters are assumed to be located at the center of

a a a a the waveguide cross section, i.e., ( 2 , 2 , ri) and ( 2 , 2 , rj). In this case, the mode function for TE10 q ωkz ~ ikz(r−ok ) 2 mode is given by uk (ri) = yeˆ z with S = a . By reducing the cross section, we z 0LS can increase the amplitude of the mode function and therefore the coupling strength. are shown in

33 Fig. 3.2(b). To simplify the problem, we assume that the transtion dipole moment of the emitter √ is along the y direction and the size of the waveguide satisﬁes λ0/2 < a < λ0/ 2 where λ0 =

2πc/ω0 with ω0 being the transition frequency of the emitter. In this case, the emitter is mainly coupled to the TE10 mode (Fig. 3.2(b)). The density of states of EM ﬁeld in the waveguide is D(ν) = L √ ν TE πc2 ν 2 π 2 . The coupling strength between the emitter and the 10 mode is therefore ( c ) −( a ) p given by g ≡ µ · E/~ = µ ν/0LS~ [46]. The single emitter decay rate due to the waveguide modes is

2 2 X 2 2µ ω0 γ1d = 2π |g(ν)| δ(ω0 − ν) = ≡ ηγ0, (3.32) Sc2k ν ~ 0 0z

2 where η = 3λ0λ0z/(2πa ) is the enhancement factor, λ0z = 2π/k0z is the effective longitudinal

wavelength and γ0 is the spontaneous decay rate in the free space. Around the cutoff frequency,

we have k0z → 0 and therefore η → ∞, i.e., the spontaneous decay rate can be greatly enhanced. The dispersion relations Then we need to derive the master equation in the waveguide case. Compared with the free space case, the only modiﬁcation to the calculation for the waveguide is P → P in Eq.(3.13). ks kz We here calculate the ﬁrst and the second term in Eq.(3.13) as an example. For the second term,

34 we have

Z t X + iω0t ∗ ∗ − −iω0(t−τ) −iωk0s0 τ 2 S − dτµi · uks(ri)Si e µj · uk0s0 (rj)Sj e e cosh rρ (t − τ)δkk0 δss0 0 kz Z ∞ Z t 2 L ωkµ iω0τ −iωkz τ ikz(ri−rj ) 2 + − S = − dkz dτe e e cosh rSi Sj ρ (t − τ) 2π −∞ 0 0LS~ ∞ t 2 L Z Z 2 ω µ iω0τ −i[ω0+c k0z(kz−k0z)/ω0]τ k ikz(ri−rj ) −ikz(ri−rj ) 2 + − S ≈ − dkz dτe e [e + e ] cosh rSi Sj ρ (t − τ) 2π 0 0 0LS~ ∞ t 2 L Z Z 2 ω µ −iτc k0zδkz/ω0 k i(k0z+δkz)(ri−rj ) −i(k0z+δkz)(ri−rj ) 2 + − S ≈ − dδkz dτe [e + e ] cosh rSi Sj ρ (t − τ) 2π −k0z 0 0LS~ ∞ t 2 L Z Z 2 ω µ −i(c k0zδkz/ω0)τ k i(k0z+δkz)(ri−rj ) −i(k0z+δkz)(ri−rj ) 2 + − S ≈ − dδkz dτe [e + e ] cosh rSi Sj ρ (t − τ) 2π −∞ 0 0LS~ L Z t ω µ2 c2k c2k 0 ik0z(ri−rj ) 0z −ik0z(ri−rj ) 0z ≈ − dτ 2π[e δ((ri − rj) − τ) + e δ((ri − rj) + τ)] 2π 0 0LS~ ω0 ω0 2 + − S × cosh rSi Sj ρ (t − τ) L ω µ2 ω ik0zrij 0 0 2 + − S ≈ − e 2π 2 cosh rSi Sj ρ (t) 2π 0LS~ c k0z γ γ ≈ − [ 1d cos(k r ) + i 1d sin(k r )] cosh2 rS+S−ρS(t) 2 0z ij 2 0z ij i j γ ≡ − ( ij + iΛ ) cosh2 rS+S−ρS(t) 2 ij i j (3.33)

2 2 2 where emitter separation rij = |ri − rj|, γ1d = 2µ ω0/~0Sc k0z is the spontaneous decay rate

in the waveguide, γij = γ1d cos(k0zrij) is the collective decay rate, and Λij = γ1d sin(k0zrij)/2

p π 2 2 is the collective energy shift. In the third line we expand ωk = c ( a ) + (kz) around kz = k0z since resonant modes provide dominant contributions. In the ﬁfth line we extend the integration R ∞ R ∞ dkz → dkz because the main contribution comes from the components around δkz = 0. −k0z −∞

In the next line, Weisskopf-Wigner approximation is used. Thus, we have obtained γij and Λij.

35 Next we need to calculate the ﬁrst term (squeezing term) in Eq.(3.13):

Z t X + + i(ωk−ω0)τ S dτ{µi · u2k0−k(ri)Si µj · uk(rj)Sj e [− sinh(r) cosh(r)]ρ (t − τ) 0 kz Z 2k0z Z t L i(ω −ω )τ i(2k −k )(r −o ) ik (r −o ) = − dk dτe kz 0 e 0z z i 1 e z j 1 2π z √0 0 ω ω µ2 kz 2k0z−kz + + S (3.34) × sinh(r) cosh(r)Si Sj ρ (t − τ) 0LS~ L Z 0 Z t i(ωk −ω0)τ i(−2k0z−kz)(ri−o2) ikz(rj −o2) − dkz dτe z e e 2π −2k0z 0 √ 2 ωkz ω−2k0z−kz µ + + S × sinh(r) cosh(r)Si Sj ρ (t − τ) 0LS~

For i = j, Eq.(5.8) reduces to

Z t X + + i(ωk−ω0)τ S dτ{µi · u2k0−k(ri)Si µj · uk(rj)Sj e [− sinh(r) cosh(r)]ρ (t − τ) 0 kz 2k t 2 √ 2 Z 0z Z c k0z L i (kz−k0z)τ ωkz ω2k0z−kz µ ω i2k0z(ri−o1) + + S = − dkz dτe 0 e sinh(r) cosh(r)Si Sj ρ (t − τ) 2π 0 0 0LS~ 0 t 2 √ 2 Z Z c k0z L i (kz−k0z)τ ωkz ω−2k0z−kz µ ω −i2k0z(ri−o2) + + S − dkz dτe 0 e sinh(r) cosh(r)Si Sj ρ (t − τ) 2π −2k0z 0 0LS~ L ω µ2 Z t c2k i2k0z(ri−o1) −i2k0z(ri−o2) k0z 0z + + S = − [e + e ] dτ2πδ( τ) sinh(r) cosh(r)Si Sj ρ (t − τ) 2π 0LS~ 0 ω0 L ω µ2 Z t c2k i2k0z(ri−o1) −i2k0z(ri−o2) k0z 0z + + S = − [e + e ] dτ2πδ( τ) sinh(r) cosh(r)Si Sj ρ (t − τ) 2π 0LS~ 0 ω0 ω2µ2 i2k0zR 0 + + S = −e 2 cos(2k0zri) sinh(r) cosh(r)Si Sj ρ (t) 0~Sc k0z γ1d = −ei2k0zR cos(2k r ) sinh(r) cosh(r)S+S+ρS(t) 2 0z i i j (3.35)

where we have used the fact that the origin of coordinate system is at equal distant from two

0 sources(i.e., o2 = −o1 = R) in the second last line. Thus, we have γii = γ1d cos(2k0zri). For

36 i 6= j, Eq. (5.8) reduces to

Z t X + + i(ωk−ω0)τ S dτ{µi · u2k0−k(ri)Si µj · uk(rj)Sj e [− sinh(r) cosh(r)]ρ (t − τ) 0 kz 2k t 2 Z 0z Z c k0z L i (kz−k0z)τ ω i2k0z(rc−o1) −i(kz−k0z)(ri−rj ) = − dkz dτe 0 e e 2π 0 0 √ 2 ωkz ω2k0z−kz µ + + S × sinh(r) cosh(r)Si Sj ρ (t − τ) 0LS~ 0 t 2 Z Z c k0z L i (−kz−k0z)τ ω −i2k0z(rc−o2) −i(kz+k0z)(ri−rj ) − dkz dτe 0 e e 2π −2k0z 0 √ 2 ωkz ω−2k0z−kz µ + + S × sinh(r) cosh(r)Si Sj ρ (t − τ) 0LS~ 2k t 2 Z 0z Z c k0z L i (kz−k0z)τ ω i2k0z(rc−o1) −i(kz−k0z)(ri−rj ) = − dkz dτe 0 e e 2π 0 0 √ 2 ωkz ω2k0z−kz µ + + S × sinh(r) cosh(r)Si Sj ρ (t − τ) 0LS~ 2k t 2 Z 0z Z c k0z L i (kz−k0z)τ ω −i2k0z(rc−o2) −i(−kz+k0z)(ri−rj ) − dkz dτe 0 e e 2π 0 0 √ 2 ω−kz ω−2k0z+kz µ + + S × sinh(r) cosh(r)Si Sj ρ (t − τ) 0LS~ ∞ t 2 2 Z Z c k0z L ω µ i (kz−k0z)τ i2k0z(rc−o1) k0z ω −i(kz−k0z)(ri−rj ) = − e dkz dτe 0 e 2π 0LS~ −∞ 0 + + S × sinh(r) cosh(r)Si Sj ρ (t − τ) ∞ t 2 2 Z Z c k0z L ω µ i (kz−k0z)τ −i2k0z(rc−o2) k0z ω i(kz−k0z)(ri−rj ) − e dkz dτe 0 e 2π 0LS~ −∞ 0 + + S × sinh(r) cosh(r)Si Sj ρ (t − τ) L ω µ2 Z t c2k c2k i2k0zR 0 i2k0zrc 0z −i2k0zrc 0z = − e dτ2π[e δ(ri − rj − τ) + e δ(ri − rj + τ)] 2π 0LS~ 0 ω0 ω0 + + S × sinh(r) cosh(r)Si Sj ρ (t − τ) ω2µ2 γ i2k0zR 0 i2k0zrcsgn(i−j) + + S 1d i2k0zR + + S = − e 2 e Si Sj ρ (t) → − e cos(k0z(ri + rj))Si Sj ρ (t) 0~Sc k0z 2 (3.36) where sgn(i − j) is the sign function. The last arrow is because we need to sum over i, j, so the

i2k0zrcsgn(i−j) 0 i2k0zR imaginary part of e vanishes and the neat result is that γij = e γ1d cos(k0z(ri +

37 + S + rj)). As for Si ρ (t)Sj terms, the combination of the last two terms in Eq.(3.11) will make the

i2k0zrcsgn(i−j) 0 i2k0zR imaginary part of e vanish. Thus, we have γij = e γ1d cos(k0z(ri + rj)). If one

0 needs to get γij, γij and Λijin the unidirectional waveguide case, we just need to discard the second terms in the parenthesis of Eq.(5.7) and Eq.(5.10). In conclusion, the master equation in the 1D waveguide is also given by Eq. (3.25), but the coefﬁcients are replaced by:

γij = γ1d cos(k0zrij) γ Λ = 1d sin(k r ) (3.37) ij 2 0z ij 0 γij = γ1d cos[k0z(ri + rj)]

p ω0 2 cπ 2 where k0z = ( c ) − ( a ) is the wave vector along the waveguide direction and rij = |ri − rj| is the separation between two emitters. It is worth noting that Eq. (3.25) is valid not only for the rectangular waveguide, but also for arbitrary type of waveguide with arbitrary atomic transition frequency. The only difference for different types of waveguide and different transition frequency is the value of γ1d in Eq. (5.13). Similar to the 3D case, the two-photon decay rate induced by the squeezed vacuum depends on the center of mass of the emitters. This can be explained by the interference shown in Fig. 3.2(b). The emitters can absorb two photons from the squeezing sources either from the left or the right. These two processes can interfere with each others and we have

0 1 2 1 2 2ik0zR γij ∝ SLSL + SRSR = 2e cos[k0z(ri + rj)]

which is a periodic function with period λ0z. Thus, when the center of mass happens to be at the antinodes (nodes) of the standing wave, the two-photon decay rate is maximized (minimized).

38 (a)

1.0

0.8

0.6 (t) x,y

0.4

x,y

0.2

0.0

0.0 0.5 1.0 1.5 2.0

(b)

1.4

1.2

1.0 1d

0.8 /

dp 0.6

0.4

0.2

x y

0.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 3.3: (a) The dephasing dynamics of a single emitter in the squeezed vacuum. The black and red solid curves are the results of σx and σy, respectively. The blue dotted line is the result when there is no squeezing (thermal reservoir). (b) The dephasing rates of σx and σy as a function of the emitter position. For (a)&(b), the squeezing parameters are chosen to be r = 0.5.

39 3.3 Dynamics of a single qubit

In this section, we will show the difference between the traditional theory and our theory when applied to the single two-level atom case. We still assume that the atom is located at (0, 0, δ), with the transition dipole moment along the y-axis. By eliminating the terms with i 6= j, the master equation shown in Eq.(3.25) is reduced to the single-atom case which is given by

dρS = sinh(r) cosh(r)γ0(e2ik0zRS+ρSS+ + H.c.) dt 1 − γ cosh2(r)(ρSS+S− + S+S−ρS − 2S−ρSS+) (3.38) 2 1 − γ sinh2(r)(ρSS−S+ + S−S+ρS − 2S+ρSS−) 2

0 + S + with γ = γ1d and γ = γ1d cos(2k0δ). It is worth noting that the squeezing terms like S ρ S and S−ρSS− in Eq. (5.15) only affect the non-diagonal terms but not the diagonal terms. Thus, for single emitter, the squeezing can only modify the dephasing rate rather than the population decay rate. We also notice that the dephasing rate due to the squeezed vacuum is dependent on the emitter position because the interference between the two squeezing sources generates a standing wave.

The dynamical equations for the expectation value of σ+ and σ− are given by

    d hσ+i hσ+i   = U   (3.39) dt     hσ−i hσ−i

where

  −(N + 1 ) Me−2ik0zR cos(2k δ)  2 0z  U = γ1d   . (3.40) 2ik0zR 1 Me cos(2k0zδ) −(N + 2 )

1 The eigenvalues of U are γdp,± = [N + 2 ± M cos(2k0zδ)]γ1d which are the dephasing rate. In fact, such a position-dependent property of the dephasing rate can be associated with the variance in the quadrature phases of the squeezed ﬁeld at the site of the atom. Considering

40 the operator X(δ, α, β) = √1 (ei(k0z+kz)δa eiα + ei(k0z−kz)δa eiβ + H.c.) which de- 2 2 k0z+kz k0z−kz scribes the entangled modes of the two-mode squeezing, we can ﬁnd its variance ∆X(δ, α, β) =

1 1 2 [N + 2 −M cos(2k0zδ +α+β)]. Therefore, we have the relation that γdp,+ = 2∆X(δ, α+β = 0)

and γdp,− = 2∆X(δ, α + β = π).

We can see that when there is no squeezing, i.e., M = 0, both σx and σy have the same

2 dephasing rate cosh (r)γ1d/2 (blue dotted line in Fig. 3.3(a)). However, if there is squeezing, i.e.,

M 6= 0, σx and σy have different dephasing rates with one being enhanced and the other one being suppressed (solid lines in Fig. 3.3(a)). The dephasing rate can be tuned by changing the position

of the emitter. In Fig. 3.3(b), it is shown that the dephasing rates of σx and σy vary periodically as

the emitter position changes. At some regions, σx decays faster than σy, while at other regions, σx

decays slower than σy. This result challenges the traditional conclusion where dephasing rate is a position-independent constant[4, 30]. The power spectrum of the resonance ﬂuorescence can also be calculated and the result is similar to Ref. [32] with the simple replacements of M by Mγ0 and the phase of M by e2ik0zR.

3.4 Dynamics of multiple qubits coupled by the dipole-dipole interaction

Next, we consider the two-emitter case where dipole-dipole interaction can occur and two-

photon process is allowed. In Fig. 3.4(a), we show the dynamics of the transverse polarization σx

and σy. Here, we compare two different emitter separations r12 = 0.5λ0z and r12 = 1.0λ0z. In both cases, the x and y polarizations have the same decay dynamics in the thermal reservoir. However, in the squeezed vacuum, the two orthogonal polarizations have different decay rates with one being enhanced and the other being suppressed. When r12 = 0.5λ0z, σx decays faster than that in the thermal reservoir, but σy decays much slower than that in the thermal reservoir. While opposite result occurs when r12 = 1.0λ0z. This is similar to the one-emitter case. Different from the one-emitter case, as is shown in Fig. 3.4(b), the squeezed vacuum can affect the population decay of the two-emitter system. This is because two-photon process is allowed in the two-emitter system. Without the squeezed vacuum, the system is ﬁnally in the thermal equilibrium state (dotted lines). However, the squeezed vacuum can deplete the populations on

41 (b) (a) 1.0 1.0

(1)

0.8 0.8 y

(2)

0.6 0.6 (t)

x x,y

0.4 0.4

(2) (1) Population

(1)

x(y) x(y)

x --

++ 0.2 0.2

(2)

0.0 0.0

0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10

t t

(c)

1.4

y (d) x

2.0

1.2

y x

1.8

1.0

1.6 1d

0.8 d / 1.4 dp / 0.6

1.2 dp

0.4 1.0

0.8 0.2

y x

0.6

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.5 1.0 1.5

r /

c 0z

12 0z

Figure 3.4: Two-emitter case: Transverse polarization decay of the first emitter as a function of time. (a) r12 = 0.5λ0z for superscript (1) and r12 = 1.0λ0z for superscript (2), r = 0.5 and rc = 0; (b) population decay as a function of time when r12 = 0.5λ0z, r = 0.5 and rc = 0. Solid lines are the results in squeezed vacuum and the dotted lines are the results in the thermal reservoir with 2 N = sinh (r). Here the dynamics of ρ++ and ρ−− are highly identical. (c) Dephasing rate as a function of atom separation with the center of mass fixed at rc = 0. (d) Dephasing rate as a function of center of mass position with atom separation fixed at rij = λ0z, where the two-atom case is plotted in solid lines and the five-atom case is plotted in dashed lines.

42 | + +i and | − −i with |±i = √1 (|e i|g i ± |g i|e i). In fact, the atomic pair evolves into an 2 1 2 1 2 entanglement state in this case and we will discuss it later. We also study the dephasing rate as a function of emitter separation and position of the center of mass which are shown in Fig. 3.4(c) and (d) respectively. Here the dephasing rate is defined to be the inverse of time for σx(σy) to damp to 1/e of its initial value. Similar to the one-emitter case, the dephasing rate is a periodic function of both r12 and rc. However, due to the dipole- dipole interaction, the dephasing rate is no longer a constant even in the thermal reservoir (dotted line in Fig. 3.4(c)) so that the value ranges of σx and σy are no longer the same in the squeezed vacuum(solid lines in Fig. 3.4(c)). It is noted that when r12 = 0.5nλ0z (n is any integer) σy does not decay to 1/e of its initial value due to the subradiance effect. When we fix the atom separation and change the center of mass(Fig. 3.4(d)), the dephasing rate changes periodically and harmonically like one-emitter case. Therefore, the dephasing rate is tunable by changing the atom separation or position of center of mass. Usually, the positions of the atoms are not easy to be tuned. However, we can easily tune the position of the squeezing sources to effectively change the center of mass of the atoms. Figure 3.4(d) also shows the result when there are five emitters (dashed lines). The dephasing rate is significantly increased when Na increases due to the collective effect, which depends on the number of atoms but not its parity.

3.5 Generalization to multi-level atoms

In this section, we will consider a more general case for atoms of arbitrary number of energy levels in the squeezed vacuum reservoir. For the rectangular waveguide, there is no TM01 or TM10 mode. Assuming b < a, TE10 is the ground mode with the lowest cutoff frequency. For simplicity, we assume that all electronic transitions only coupled to TE10 mode. The atom-ﬁeld system is described by the Hamiltonian

H = HA + HF + HAF (3.41)

PNa P where HA = l=1 e=a,b,c ~ωe,l |eli hel| is the atomic Hamiltonian, and |eli is the energy state of

43 P † 1 the lth atom with energy ~ωe,l. The Hamiltonian of the EM field is HF = ks ~ωks(âksaˆks + 2 ) † where aˆks and aˆks are the annihilation and creation operators of the filed mode with wavevector k, polarization s (in waveguide, it represents TEmn or TMmn), and frequency ωk,s. The inter-

P P PNa action Hamiltonian in the electric-dipole approximation is HAF = −i~ ks i=1,2 l=1[µl,i ·

+ ∗ − uks(rl,i)Sl,iaˆks + µl,i · uks(rl,i)Sl,iaˆks − H.c.] where µl,i is the electric dipole moment for the ith transition of the lth atom, where i = 1 denotes the transition from |ai to |bi, and i = 2 denotes the

+ − transition from |bi to |ci. Here, Sl,i and Sl,i are the raising and lowering operator for the transition

i of the lth atom. The mode function of the squeezed vacuum in the TE10 mode is given by

r ω ks ik·(ri−oks) uks(ri) = xe (3.42) 20~V

Thus, the interaction Hamiltonian is where

X † ∗ − D(t) = [µl,i · uk,s(rl,i)Sl,i(t) + µl,i · uk,s(rl,i)Sl,i(t)] (3.43) l,i

The reduced master equation of atoms in the reservoir is:

S Z t dρ 1 S F = − 2 dτT rF {[V (t), [V (t − τ), ρ (t − τ)ρ } dt ~ 0 Z t 1 S F S F (3.44) = − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ + ρ (t − τ)ρ V (t − τ)V (t) ~ 0 − V (t)ρS(t − τ)ρF V (t − τ) − V (t − τ)ρS(t − τ)ρF V (t)}

Here we just show how to deal with the ﬁrst term in Eq. (3.11), the remaining terms can be

44 calculated in the same way. For the ﬁrst term, we have

Z t 1 S F − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ } ~ 0 Z t X F + F † = dτ {D(t)D(t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] − D(t)D (t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] 0 ks,k0s0

+ F † + + F † † S − D (t)D(t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] + D (t)D (t − τ)T rF [ρ aks(t)ak0s0 (t − τ)]}ρ (t − τ)}. (3.45)

Under the rotating wave approximation(RWA), we have

Z t 1 S F − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ } ~ 0 Z t X X + iωit + iωj (t−τ) = dτ{µl,i · uks(rl,i)Sl,ie µm,j · uk0s0 (rm,j)Sm,je ijlm ks,k0s0 0

−i(ωks+ω 0 0 )t+iω 0 0 τ k s k s 0 0 × e [−Mδk ,2k0−kδss ]

+ iωit ∗ ∗ − −iωj (t−τ) −iωk0s0 τ 2 − µl,i · uks(rl,i)Sl,ie µm,j · uk0s0 (rm,j)Sm,je e cosh rδkk0 δss0 (3.46) ∗ − −iωit ∗ + iωj (t−τ) −iωk0s0 τ 2 − µl,i · uks(rl,i)Sl,ie µm,j · uk0s0 (rm,j)Sm,je e cosh rδkk0 δss0

∗ ∗ − −iωit + iωj (t−τ) iωk0s0 τ 2 − µl,i · uks(rl,i)Sl,ie µm,j · uk0s0 (rm,j)Sm,je e sinh rδkk0 δss0

∗ + iωit ∗ − −iωj (t−τ) iωk0s0 τ 2 − µl,i · uks(rl,i)Sl,ie µm,j · uk0s0 (rm,j)Sm,je e sinh rδkk0 δss0

∗ ∗ − −iωit ∗ ∗ − −iωj (t−τ) i(ωks+ωk0s0 )t−iωk0s0 τ + µl,i · uks(rl,i)Sl,ie µm,j · uk0s0 (rm,j)Sm,je e

S 0 0 × [−Mδk ,2k0−kδss ]}ρ (t − τ) where l, m are used for labeling different atoms, and i, j are used for transitions within an atom.

Here we just calculate the ﬁrst and second term. Since all atoms are identical, ωl,i = ωi, |µl,i| =

|µi|, and rl,i = rl can be used to simplify Eq. (3.46). For simplicity, we deﬁne µj to be the

45 projection of µj on the x axis. For the second term(thermal term), we have

Z t X + iωit ∗ ∗ − −iωj (t−τ) −iωk0s0 τ 2 S − dτµl,i · uks(rl)Sl,ie µm,j · uk0s0 (rm)Sm,je e cosh rρ (t − τ)δkk0 δss0 0 kz Z ∞ Z t L ωkµiµj i(ωi−ωj )t iωj τ −iωkz τ ikz(rl−rm) 2 + − S = − e dkz dτe e e cosh rSl,iSm,jρ (t − τ) 2π −∞ 0 0LS~ ∞ t L Z Z 2 ω µ µ i(ωi−ωj )t iωj τ −i[ωj +c kjz(kz−kjz)/ωj ]τ k i j ≈ − e dkz dτe e 2π 0 0 0LS~

ikz(rl−rm) −ikz(rl−rm) 2 + − S × [e + e ] cosh rSl,iSm,jρ (t − τ) ∞ t L Z Z 2 ω µ µ i(ωi−ωj )t −iτc kjzδkz/ωj k i j ≈ − e dδkz dτe 2π −k0z 0 0LS~

i(kjz+δkz)(rl−rm) −i(kjz+δkz)(rl−rm) 2 + − S × [e + e ] cosh rSl,iSm,jρ (t − τ) ∞ t L Z Z 2 ω µ µ i(ωi−ωj )t −i(c kjzδkz/ωj )τ k i j ≈ − e dδkz dτe 2π −∞ 0 0LS~

i(kjz+δkz)(rl−rm) −i(kjz+δkz)(rl−rm) 2 + − S × [e + e ] cosh rSl,iSm,jρ (t − τ) L Z t ω µ µ c2k c2k i(ωi−ωj )t j i j ikjz(rl−rm) jz −ikjz(rl−rm) jz ≈ − e dτ 2π[e δ((rl − rm) − τ) + e δ((rl − rm) + τ)] 2π 0 0LS~ ω0 ω0 2 + − S × cosh rSl,iSm,jρ (t − τ)

L ωjµiµj ωj ≈ − eikjzrlm 2π cosh2 rS+ S− ρS(t)ei(ωi−ωj )t 2π LS c2k l,i m,j √ 0 ~ √0z γiγj γiγj ≈ − [ cos(k r ) + i sin(k r )] cosh2 rS+ S− ρS(t)ei(ωi−ωj )t 2 0z lm 2 0z lm l,i m,j (3.47)

2 2 2 where emitter separation rlm = |rl − rm|, collective decay rate γi = 2µi ωi /~0Sc kiz, and col-

√ p π 2 2 lective energy shift Λij = γiγj sin(k0zrij)/2. In the third line we expand ωk = c ( a ) + (kz)

around kz = k0z since resonant modes provide dominant contributions. In the ﬁfth line we extend R ∞ R ∞ the integration dkz → dkz because the main contribution comes from the components −kjz −∞

around δkz = 0. In the next line, Weisskopf-Wigner approximation is used. Next we need to calculate the ﬁrst term (squeezing term) in Eq. (3.46), putting aside the overall

46 factor ei(ωi+ωj −2ω0)t, we have:

Z t X + + i(ωk−ωj )τ S dτ{µl,i · u2k0−k(rl)Sl,iµm,j · uk(rm)Sm,je (−M)ρ (t − τ) k 0 z √ Z 2k0z Z t L ωkz ω2k0z−kz µiµj i(ωkz −ωj )τ i(2kjz−kz)(rl−o1) ikz(rm−o1) + + S = − dkz dτe e e MSl,iSm,jρ (t − τ) 2π 0 0 0LS~ Z 0 Z t √ L ωkz ω−2k0z−kz µiµj i(ωkz −ωj )τ i(−2kjz−kz)(rl−o2) ikz(rm−o2) + + S − dkz dτe e e MSl,iSm,jρ (t − τ) 2π −2k0z 0 0LS~ (3.48)

For terms with rl = rj, Eq. (5.8) reduces to

Z t X + + i(ωk−ωj )τ S dτ{µl,i · u2k0−k(rl)Sl,iµl,j · uk(rl)Sl,je (−M)ρ (t − τ) 0 kz 2 √ Z 2k0z Z t c kjz L i (kz−kjz)τ ωkz ω2k0z−kz µiµj ωj i2k0z(rl−o1) + + S = − dkz dτe e MSl,iSl,jρ (t − τ) 2π 0 0 0LS~ 2 √ Z 0 Z t c kjz L i (kz−kjz)τ ωkz ω−2k0z−kz µiµj ωj −i2k0z(rl−o2) + + S − dkz dτe e MSl,iSl,jρ (t − τ) 2π −2k0z 0 0LS~ √ t 2 L ωiωjµiµj Z c k i2k0z(rl−o1) −i2k0z(rl−o2) jz + + S = − [e + e ] dτ2πδ( τ)MSl,iSl,jρ (t − τ) 2π 0LS~ 0 ωj 2 ω µiµj = − ei2kjzR 0 cos(2k r )MS+ S+ ρS(t) Sc2k 0z l l,i l,j √0~ 0z γiγj = − ei2k0zR cos(2k r )MS+ S+ ρS(t) 2 0z l l,i l,j (3.49) where we have used the fact that the origin of coordinate system is at equal distant from two

0 sources(i.e., o2 = −o1 = R) in the second last line. Incorporating index l into i, we have γij =

47 √ γiγj cos(2k0zri). For ri 6= rj, Eq. (5.8) reduces to

Z t X + + i(ωk−ωj )τ S dτ{µl,i · u2k0−k(rl)Sl,iµm,j · uk(rm)Sm,je (−M)ρ (t − τ) 0 kz 2 Z 2k0z Z t c kjz L i (kz−kjz)τ i2k (r −o ) −i(k −k )(r −r ) = − dk dτe ωj e 0z c 1 e z 0z l m 2π z √0 0 ωkz ω2k0z−kz µiµj + + S × MSl,iSm,jρ (t − τ) 0LS~ 2 Z 0 Z t c kjz L i (−kz−kjz)τ −i2k (r −o ) −i(k +k )(r −r ) − dk dτe ωj e 0z c 2 e z 0z l m 2π z √−2k0z 0 ωkz ω−2k0z−kz µiµj + + S × MSl,iSm,jρ (t − τ) 0LS~ √ 2 Z ∞ Z t c kjz L ωiωjµiµj i (kz−kjz)τ i2k0z(rc−o1) ω −i(kz−k0z)(rl−rm) = − e dkz dτe j e 2π 0LS~ −∞ 0 + + S × MSl,iSm,jρ (t − τ) √ 2 Z ∞ Z t c kjz L ωiωjµiµj i (kz−kjz)τ −i2k0z(rc−o2) ω i(kz−k0z)(rl−rm) − e dkz dτe j e 2π 0LS~ −∞ 0 + + S × MSl,iSm,jρ (t − τ) L ω2µ µ Z t c2k c2k i2k0zR 0 i j i2k0zrc 0z −i2k0zrc 0z ≈ − e dτ2π[e δ(rl − rm − τ) + e δ(rl − rm + τ)] 2π 0LS~ 0 ω0 ω0 + + S × MSl,iSm,jρ (t − τ) 2 ω µiµj ≈ − ei2k0zR 0 ei2k0zrcsgn(rl−rm)S+ S+ ρS(t) Sc2k l,i m,j √ 0~ 0z γiγj → − ei2k0zR cos(k (r + r ))S+ S+ ρS(t) 2 0z l m l,i m,j (3.50)

where sgn(rl − rm) is the sign function. The last arrow is because we need to sum over i, j, so the

i2k0zrcsgn(i−j) 0 i2k0zR√ imaginary part of e vanishes, so the neat result is that γijkl = e γjγl cos(k0z(ri+

+ S + rk)). As for Si ρ (t)Sj terms, the combination of the last two terms in Eq. (3.11) makes the

imaginary part of ei2k0zrcsgn(rl−rm) vanish. Doing similar calculations for the remaining terms, we

48 have the master equation

dρS X = − i Λ [S+ S− , ρS]ei(ωj −ωl)t dt ijkl i,j k.l ijkl 1X − γ (1 + N)(ρSS+ S− + S+ S− ρS − 2S− ρSS+ )ei(ωj −ωl)t 2 ijkl i,j k.l i,j k.l k.l i,j ijkl (3.51) 1X − γ N(ρSS− S+ + S− S+ ρS − 2S+ ρSS− )e−i(ωj −ωl)t 2 ijkl i,j k.l i,j k,l k,l i,j ijkl 1 X X − γ0 Me2αik0zReiα(ωj +ωl−2ω0)t(ρSSα Sα + Sα Sα ρS − 2Sα ρSSα ) 2 ijkl i,j k.l i,j k,l k,l i,j α=± ijkl with

√ γijkl = γjγl cos(k0zrik) √ γjγl Λ = sin(k r ) (3.52) ijkl 2 0z ik 0 √ γijkl = γjγl cos[k0z(ri + rk)]

The above equation reduces to Eq. (3.25) when l = 0 indicating that the atom can be treated as a qubit.

49 4. THE STEADY STATE PROPERTIES OF ATOMS IN THE SQUEEZED VACUUM1

It is worth noting that in our scheme, the squeezed vacuum reservoir is not affected by the atomic behaviors because of there are a great number of modes in the reservoir. What’s more, the squeezed vacuum is pumped into the waveguide endlessly so that the effects left by the atoms will be erased eventually. Therefore, the behaviors of the steady state of the atoms in the squeezed vacuum are expected to be quite different from those in the ordinary vacuum or thermal reservoir. In this section, we will study the properties of the steady state with different atomic structures.

4.1 Quantum entanglement of two qubits

Quantum entanglement is an important resource of the quantum information and quantum metrology [47, 48]. Preparation of the maximum entangled state is still a central topic of inter- est. It has been shown that stationary quantum entanglement can be dissipatively prepared by engineering the bath enviroment [49, 50, 51, 52]. By squeezing the enviroment, quantum entanglement between emitters can be also created [53, 34, 35]. However, it is shown in Ref. [34] that stationary maximum entanglement can not be reached by the squeezed vacuum for identical emitters. Here, we show that identical emitters coupled to the 1D waveguide can also be driven to a stationary maximum entangled NOON state by the squeezed vacuum as long as the center of mass is put at the proper position. The quantum entanglement can be measured by the concurrence which is deﬁned as [54]:

C ≡ max{0, λ1 − λ2 − λ3 − λ4} in which λ1, λ2, λ3, λ4 are eigenvalues, in decreasing order, of

p√ √ N ∗ N the Hermitian matrix R = ρρe ρ with ρe = (σy σy)ρ (σy σy). For a pure two-qubit state |Ψi = α|eei + β|egi + γ|gei + |ggi with |α|2 + |β|2 + |γ|2 + |δ|2 = 1, the concurrence is given by C = max{0, 2|αδ − βγ|}. According to the master equation Eq. (3.25) describing qubits in the waveguide, the concurrence of two qubits as a function of time for different initial states can

50 be calculated, which is shown in Fig. 4.1(a) where r = 1, rc = 0, and r12 = 0.25λ0z. Different curves correspond to different initial states. We can see that no matter what the initial state is, the two-emitter state will be driven to a very high entangled state. To see what the stationary state is, we also show the ﬁdelity of the emitter state with respect to the maximum entangled state

√1 (|ggi − |eei) which is shown in Fig. 4.1(b). We can see that the stationary state is very close 2 to it. Therefore, under these parameters the two emitters can be driven to the maximum entangled state which may ﬁnd important applications in quantum information and quantum computation. To ﬁnd the stationary state analytically, we rewrite the master equation in Eq. (3.25) as

ρ˙gg = −2Nγρgg + (N + 1)γ+ρ++ + (N + 1)γ−ρ−−

0 +Mγ12ρu, (4.1)

ρ˙ee = −2(N + 1)γρee + Nγ+ρ++ + Nγ−ρ−−

0 +Mγ12ρu, (4.2)

ρ˙++ = −(2N + 1)γ+ρ++ + (N + 1)γ+ρee + Nγ+ρgg

0 −Mγ+ρu, (4.3)

ρ˙−− = −(2N + 1)γ−ρ−− + (N + 1)γ−ρee + Nγ−ρgg

0 −Mγ−ρu. (4.4)

0 0 ρ˙u = −(2N + 1)γ11ρu − 2Mγ+ρ++ − 2Mγ−ρ−−

0 +2Mγ12(ρee + ρgg). (4.5)

where ρ = hee|ρ|eei, ρ = hgg|ρ|ggi, ρ = h±|ρ|±i with |±i = √1 (|e i|g i ± |g i|e i), ee gg ±± 2 1 2 1 2 0 −2ik0zR 2ik0zR ρu = e hee|ρ|ggi + e hgg|ρ|eei, and γ = γ1d, γ± = γ1d(1 ± cos(k0zr12)), γ12 =

0 1 (r1+r2) γ1d cos(2k0zrc), γ± = γ1d{cos[2k0zrc] ± 2 [cos(2k0zr1) + cos(2k0zr2)]} with rc = 2 . Then the

51 (a)

1.0

gg 0.8

0.6

entangled

0.4 eg

0.2

ee gg concurrence 0.0

-0.2 non-entangled

-0.4

0 20 40

(b)

1.0

0.8

ee ee gg

0.6

eg fidelity

0.4

0.2

0.0

0 10 20 30 40

Figure 4.1: (a) Concurrence evolution of different initial states in squeezed vacuum, where r = 1, rc = 0, and r12 = 0.25λ0z. (b) Fidelity evolution of different initial states in the same environment.

52 steady state solutions are given by

2 2 N[−1 − N − 2N + (−1 + N + 2N ) cos(4k0zrc)] ρee = 2 2(1 + 2N)[−1 − 2N − 2N + 2N(1 + N) cos(4k0zrc)] 2 N(1 + N)sin (2k0zrc) ρ++ = − 2 −1 − 2N − 2N + 2N(1 + N) cos(4k0zrc) 2 (4.6) N(1 + N)sin (2k0zrc) ρ−− = − 2 −1 − 2N − 2N + 2N(1 + N) cos(4k0zrc) p −2 N(1 + N) cos(2k0zrc) ρu = 2 (1 + 2N)[−1 − 2N − 2N + 2N(1 + N) cos(4k0zrc)]

where we have used the relation M 2 = N(N + 1). Obviously, the population given by Eq. (5.16)

th th differs from that given by thermal reservoir: ρee(gg) = ρee(gg) + ∆ρ, ρ++(−−) = ρ++(−−) − ∆ρ 2 N(N+1) cos (2k0zrc) th N 2 th th N(N+1) with ∆ρ = 2 2 and ρ = 2 , ρ = ρ = 2 , (1+2N) (1+2N+2N −2N(1+N) cos(4k0zrc)) ee (1+2N) ++ −− (1+2N) th (1+N)2 ρgg = (1+2N)2 which obey the Boltzmann distribution. It is interesting that the steady state depends only on the center of mass but not on the separation between the two emitters. Meanwhile, it is worth noting that the dark state cannot always be reached since the ergodicity cannot be guaranteed under every condition. For example, when cos(k0zr12) = 1, |+i becomes a dark state, while it is

|−i when cos(k0zr12) = −1.

n 0 Eq. (5.16) shows that as rc gets closer to 4 λ0z, the magnitude of γ± gets closer to ±1 which leads to smaller population on |+i and |−i as well as bigger concurrence. When the position of

n the center mass rc = 4 λ0z, the steady states are given by

N + 1 ρ = , gg (1 + 2N) N ρ = , ee (1 + 2N) (4.7)

ρ++ = ρ−− = 0, 2pN(1 + N) ρ = (−1)n+1 . u (1 + 2N)

53 with the average photon number N. When N → ∞, C → 1 which is a maximum-entangled state

√1 (|ggi − |eei) ( √1 (|ggi + |eei)) with even(odd) n. 2 2 Fig. 4.2(a) shows the dependence of the stationary quantum entanglement on the photon num-

n ber and the center-of-mass position. It is clearly seen that when rc is close to 4 λ0z the system can be prepared in a high entangled state, while the entanglement can never be formed when

2n+1 0 rc = 8 λ0z because the dipole-dipole interaction γ12 vanishes. In experiments, the center of mass position of emitters may be hard to control, but it can be effectively controllable by setting the positions squeezing sources. Thus, as long as the pump beam in SPDC is strong enough to guarantee the average photon number of the squeezed vacuum, the emitters can definitely evolve into a NOON state. While the dephasing rate is not very sensitive to the fluctuations of the emitter positions, the stationary quantum entanglement significantly depends on their center of mass. Only when the center of mass position is around nλ/4, the quantum entanglement is nonzero. In Fig. 4.2(b), we show half the range of center of mass where the quantum entanglement is non-zero. The larger the squeezing is, the more sensitive the quantum entanglement is to the fluctuation of center-of-mass. For example, when N = 1, a deviation of about 0.04λ from nλ/4 will make the entanglement vanish.

4.2 Resonance ﬂuorescence of a group of atoms

In this section, we study how the squeezing can affect the resonance fluorescence of the waveguide-QED system. In the following we study how the collective interaction, squeezing phase, squeezing degree, emitter separation, and the center of mass affect the resonance fluorescence of this system. The power spectrum of the resonance fluorescence is given by [30, 55, 56]

Z ∞ S(ω) ∝ Re dτT r[σ−(τ)σ+(0)]eiωτ . (4.8) 0

± ± ± where we assume that the detector is perpendicular to the waveguide and σ = σ1 + σ2 for the two-emitter example. The two-time correlation function in the integration can be calculated by

54 (a)

(b)

0.12

0.10

0.08 0z

0.06 / c r

0.04

0.02

0.00

0.0 0.5 1.0 1.5

Figure 4.2: (a) Concurrence of the steady state as a function of average photon number N = 2 r1+r2 sinh(r) and the position of the center mass rc = 2 .(b) The impact of rc’s ﬂuctuations on n concurrence for different average photon number N. ∆rc is the distance from 4 λ0z to the position where the entanglement vanishes. 55 1.0 1.0 (a) (b)

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 Spectrum(arb. units) Spectrum(arb. Spectrum(arb. units) Spectrum(arb.

0.0 0.0

-8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8

( ) ( )

0 0

(d) (c) 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 Spectrum(arb. units) Spectrum(arb. units) Spectrum(arb.

0.0 0.0

-8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8

( ) ( )

0 0

Figure 4.3: Resonance ﬂuorescence spectrum of the two-emitter system inside a 1D waveguide. For better comparison, the spectra are normalized to the intensity at ω = ω0 with the coherent elastic scattering singularity removed. Coherent driving Rabi frequency is ΩR = 4γ. In (a) and (b), the solid curves are the spectra for the coupled emitters, while the dashed curves are the spectra without emitter-emitter coupling. Parameters: (a) r1 = 0, r2 = 0.01λ0z, squeezing parameter r = 0.5. (b) r1 = 0, r2 = 0.25λ0z, r = 0.5. (c) r1 = 0, r2 = λ0z, φ = π/2, r = 0.5 for black line, r = 1 for red line. (d) r1 = −0.125λ0z, r2 = 0.125λ0z for the red line, r1 = −0.25λ0z, r2 = 0.25λ0z for the black line. φ = 0, r = 0.5.

56 the quantum regression theorem. Usually, the analytical result is difficult to get. However, we can resort to the numerical method to calculate the resonance fluorescence [57]. To observe the resonance fluorescence, we need to apply an external coherent driving field. The master equation is given by

dρ = −i[V, ρ] + Lρ (4.9) dt

ΩR −iα −ik0zr1 − −ik0zr2 − where Lρ is the right hand side of Eq.(3.25) and V = 2 e (e σ1 + e σ2 ) + H.c. is d·E the interaction between the driving ﬁeld and the emitters with Rabi frequency ΩR = . From ~ − − Eq. (5.19) we can evolve and obtain the steady state of the system ρss. Next we use (σ1 + σ2 )ρss as the initial condition to solve a density matrix c(t) which obeys the same equation of motion as ρ in Eq. (5.19). The resonance ﬂuorescence spectrum is then given by [57]

Z ∞ + + iωτ S(ω) ∝ Re dτT r[c(τ)(σ1 + σ2 )]e . (4.10) 0

In Fig. 4.3(a) and Fig. 4.3(b) we compare the resonance ﬂuorescence spectrum with and without the dipole-dipole interaction for different squeezing phases and emitter separations. When r12 =

0.01λ0z and φ = 0, we can see that the spectrum is very different with and without dipole-dipole interaction. Without dipole-dipole interaction, the spectrum is very similar to the typical Mollow triplet (red dashed line). However, with dipole-dipole interaction, there is a very narrow peak around the center frequency (red solid line). This is due to the subradiant state induced by the dipole-dipole interaction. On the contrary, when φ = π/2 the spectrum with and without the dipole-dipole interaction is very similar (black solid and dashed lines). From Fig. 4.3(b) we see that with dipole-dipole interaction, the spectrum can be asymmetric, i.e., the positive and negative sidebands are different. In Fig. 4.3(c) we compare the spectrum with different squeezing degrees. We can see that greater squeezing parameter leads to the power spectrum in weak-driving-ﬁeld limit(sidebands disappear). FIG. 4.3(d) shows that different emitter separation has different spectrum. This is not

57 0 only due to atomic interaction which is described by γ12, γ12, Λ12, but also due to their positions

0 which determine the values of γii, i.e., the effective phase and magnitude of M. Comparing the red solid curve in Fig. 4.3(b) and the red dashed curve in Fig. 4.3(d) we can see that different center-of-mass position can also have different resonance ﬂuorescence.

4.3 The steady state population inversion of a single Ξ-type atom by the squeezed vacuum

The concept of population inversion is of fundamental importance in laser physics because the population inversion is a key step of generating laser. However, the population inversion can never exist for a system at thermal equilibrium because of the spontaneous emission. The achievement of population inversion therefore requires pushing the system into a non-equilibrated state [31]. Thus, the spontaneous emission must be inhibited in order to maintain the population inversion in a steady state. In 1993, Ficek and Drummond studied the dynamical properties of a single three- level atom in the squeezed vacuum where they showed that a single three-level atom in the cascade conﬁguration coupled to squeezed modes in a cavity can reach steady state with level population inversion relative to the ordinary laser spectroscopy [37, 38, 39]. In their model, they found a population inversion of about 78%. Here, instead of a cavity, we consider that the case in quasi- one-dimensional waveguide. The dynamic equation can be reduced from Eq. (3.51) with ri = rk, ri = rj = 0 and the resonant condition ω1 + ω2 = 2ω0. It follows from Eq. (3.51) that various matrix elements satisfy the following equation:

1√ ρ˙ = − γ ch2ρ + γ sh2ρ − γ γ M(ρ + ρ ) (4.11a) aa 1 aa 1 bb 2 1 2 ac ca 2 2 2 2 ρ˙bb = γ1(ch ρaa − sh ρbb) + γ2(sh ρcc − ch ρbb) (4.11b) √ + γ1γ2M(ρac + ρca) 1√ ρ˙ = γ ch2ρ − γ sh2ρ − γ γ M(ρ + ρ ) (4.11c) cc 2 bb 2 cc 2 1 2 ac ca

1 2 2 <[ρ ˙ac] = − (γ1ch + γ2sh )<[ρac] 2 (4.11d) 1√ − γ γ M(ρ − 2ρ + ρ ) 2 1 2 aa bb cc

58 iδωt 1√ 2 iδωt <[e ρ˙ba] = − γ1γ2(M − 2sh )sh<[e ρbc] 2 (4.11e) 1 − ((γ + γ )ch2 + γ sh2 − γ M)<[eiδωtρ ] 2 1 2 1 1 ba

iδωt 1√ 2 −iδωt <[e ρ˙bc] = γ1γ2(2ch − M)<[(e ρab] 2 (4.11f) 1 − ((γ + γ )sh2 + γ ch2 − 2γ M)<[eiδωtρ ] 2 1 2 2 2 bc where < means real part, ch = cosh(r), sh = sinh(r), and γ1 = γab(γ2 = γbc) is the decay rate from |ai to |bi(|bi to |ci) in ordinary vacuum due to the waveguide modes. Equations (4.11e) and

(4.11f) are for the off-diagonal elements ρab, ρbc. The steady state solution of these two equations is ρab = ρbc = 0 because they are homogeneous linear equations. The ﬁrst four equations Eqs. (4.11a)-(4.11d) also have a steady state solution when they are combined with the normalization condition ρaa + ρbb + ρcc = 1. It is also worth noting that Eqs. (4.11a)-(4.11d) are independent of

δω, so the difference between ωab and ωbc does not inﬂuence the steady state of the single atom case as long as both ωab and ωbc are within the squeezing bandwidth. Thus, considering the minimum uncertainty squeezed vacuum where M = cosh(r) sinh(r), the steady state solution is:

2 sh γ2 ρaa = 2 2 , ch γ1 + sh γ2 ch2γ ρ = 1 , cc ch2γ + sh2γ 1 2 √ (4.12) chsh γ1γ2 ρac = ρca = − 2 2 , ch γ1 + sh γ2

ρbb = ρba = ρbc = 0, which is in fact a pure state of a superposition of |ai and |ci.

√ √ sh γ2 ch γ1 |ψssi = |ai − |ci. (4.13) p 2 2 p 2 2 ch γ1 + sh γ2 ch γ1 + sh γ2

Since there is no population in the state |bi, population inversion can always occur between states q |ai and |bi in the steady state. If tanh r > γ1 , population inversion can also occur between the γ2

59 state |ai and |ci. This result is similar to the result in Ref. [39]. However, in our scheme, the population inversion can approach 100% with zero population in the ground state and the middile state if γ2 γ1. In comparision, the population inversion in the cavity case shown in Ref. [39] is about of 78%.

The steady state population distribution for different ratios of γab is shown in Fig. 4.4(a). The γbc mechanism of this population inversion can be interpreted with the help of Fig. 4.5. In Fig. 4.5 we show that the direct transition between |ai, |bi, and |ci are allowed just like the thermal reservoir case. However, in the squeezed vacuum, there are additional paths for the population ﬂow: atom

in any of these three states can evolve into the other two through an intermediate “state" ρac.

Although ρac is an off-diagonal element rather than a state, it can be used to elucidate our idea.

When γab γbc, the transition rate for the |ai → |bi transition is negligible compared to γbc and √ γabγbc. Thus the atom in the state |ci can be excited to |ai through |ci → |bi → ρac → |ai, but |ai can not decay back to |ci, which results in the population trapping in the level |ai. This phenomenon is similar to the coherent population trapping, but here we achieve the trapping for Ξ structure with the squeezed vacuum reservoir, which cannot be realized with coherent pump due to spontaneous emission. Since it is hard to achieve perfect squeezing with M = pN(N + 1) in experiments, we also study the effect of different values of M on the steady state population with

1 parameters γab = 4 γbc and r = 1, which is shown in Fig. 4.4(b). In general, there is population in all three energy levels. Although the steady state population distribution is very sensitive to the value of M, the population inversion between |ai and |bi still holds for M = 0.8pN(N + 1). Only when M is larger than 0.95 can the population inversion occur between the state |ai and the state |ci.

4.4 The steady state population inversion of multiple Ξ-type atoms by the squeezed vacuum

In the last section, we demonstrated that arbitrary population inversion can occur for a single Ξ-type atom driven by the squeezed vacuum reservoir. However, with Eq. (4.12), this result can not

0 √ be simply generalized to the multi-atom case since γijij = γjγj cos[2k0zri], i.e., different atoms

0 have different γijij for the usual case unless all the atoms are perioidically distributed with period

60 1.0 (a)

0.8

0.6

0.4 Population

0.2

0.0

0.0 0.5 1.0 1.5 2.0

1/2

( / )

ab bc

(b) a

0.6

0.4 Population

0.2

0.0

0.80 0.84 0.88 0.92 0.96 1.00

M/M

ideal

Figure 4.4: (a) The steady state population distribution for different µab and µbc. The squeezing parameter r = 1 and the squeezing is perfect (M = pN(N + 1)). (b) The steady state population distribution for non-ideal squeezed vacuum which is characterized by the ratio of M and p 1 N(N + 1). The squeezing parameter r = 1, and γab = 4 γbc. 61 Figure 4.5: The allowed population ﬂow in the squeezed vacuum.

π nπ nλ/2. The squeezing term in Eq. (3.51) vanishes for atoms located around ri = + . Thus, 4k0z 2k0z for a group of randomly located atoms, if we want to achieve steady state population inversion in the squeezed vacuum, we need to modify our scheme. Here we consider the following correlation functions:

D † E 2 ak,sak0,s0 = sinh rδk0kδss0

D † E 2 ak,sak0,s0 = cosh rδk0kδss0 (4.14) D † † E −iθ 0 0 ak,sak0,s0 = −e cosh(r) sinh(r)δk ,−(2k0−k)δss

iθ 0 0 0 0 hak,sak ,s i = −e cosh(r) sinh(r)δk ,−(2k0−k)δss which indicates that the photons are entangled with those from the opposite direction. In principle, we can split the squeezed vacuum into two beams by a trianglar prism and inject them into opposite

62 ends of the waveguide. Then the coefﬁcients in the master equation shown in Eq. (3.51) become

√ γijkl = γiγk cos(k0zrjl), √ γ γ Λ = i k sin(k r ), (4.15) ijkl 2 0z jl 0 √ γijkl = γiγk cos(k0zrjl).

0 We can see that γijij is now independent of the atomic position because rjj = 0. The resulting master equation is the traditionally studied master equation for atoms in squeezed reservoir [34]. The detailed derivation of these coefficients will be derived later. Based on the master equation in Eq. (3.25) with coefficients given by above, we can show that a single atom can reach population inversion anywhere in the waveguide. When there are multiple atoms in the waveguide where the dipole-dipole interaction should be considered, our calculation shows that the population inversion can still occur for all the atoms. In fact, it is very interesting that the final state of the multiple-atom case is just the direct product of the steady state of independent atoms despite of the dipole-dipole interaction. This result can be proved by the mathematical induction.

Considering the fact that |ω1 − ω2| γi, it is reasonable to apply the secular approximation on Eq. (3.51) such that those terms with e±i(ω1−ω2)t and e±i(2ωi−2ω0)t are dropped and the master equation is then given by

dρS X = − i Λ [S+ S− , ρS] dt ijkl i,j k.j i,k,j 1X − γ (1 + N) {ρS,S+ S− } − 2S− ρSS+ 2 ijkl i,j k,j k,j i,j i,j,k (4.16) 1X − γ N {ρS,S− S+ } − 2S+ ρSS− 2 ijkl i,j k,j k,j i,j i,j,k 1 X X − γ0 M {ρS,Sα Sα } − 2Sα ρSSα 2 ijkl i,j k,l k,l i,j α=± i,k,j6=l

In the following, we use mathemtics induction to prove that steady state of this system is direct product of the steady state of a single atom. Assume that the steady state of N-atom system is ρS =

63 1.0

(a) 1.0

(b)

0.8

0.6

0.6 Fidelity Fidelity

0.4

r =

0.2 0.2

r=0.5

r =0.1

r=1

r =0.2

12 r=1.5

0.0 0.0

0 20 40 60 80 100 0 20 40 60 80 100

t/ t/

1.0

(c)

0.8

0.6 Fidelity

0.4

ab bc

0.2

ab bc

0.0

0 20 40 60 80 100

Figure 4.6: (a) Fidelity evolution with different atomic separations. The atomic separations of γ1 1 λ0, 0.1λ0, 0.2λ0 are plotted. Squeezing parameter r = 1, decay rate = and time unit √ γ2 4 τ = 1/ γabγbc is the geometric mean of the transition |ai → |bi and |bi → |ci’s spontaneous emission rates in ordinary vacuum. (b) Fidelity evolution with different squeezing parameters. De- γ1 1 cay rate = , and atomic separation r12 = λ0. (c) Fidelity evolution with different decay rates. γ2 4 Squeezing parameter r = 1, and atomic separation r12 = λ0.

64 √ √ ρ ρ ...ρ where ρ = (A|a i + C|c i)(Aha | + Chc |) and A = √ sh γ2 ,C = −√ ch γ1 . 1 2 N i i i i i 2 2 2 2 ch γ1+sh γ2 ch γ1+sh γ2 Then for (N + 1)-atom case, the extra terms induced by the (N + 1)th atom on the right hand side of Eq. (4.16) are composed of three parts: i = k = N + 1 terms, i = N + 1, k = 1, 2, ··· ,N terms, and i = 1, 2, ··· , N, k = N + 1 terms. The i = k = N + 1 terms are the exact terms for the the (N + 1)th atom as a single independent atom, so the net result of this term is 0. The terms with i = N + 1, k = 1, 2, ··· ,N are

X + − S − i ΛN+1,j,k,j[SN+1,jSk.j, ρ ] j,k 1X − γ ch2 {ρS,S+ S− } − 2S− ρSS+ 2 N+1,j,k,j N+1,j k,j k,j N+1,j j,k (4.17) 1X − γ sh2 {ρS,S− S+ } − 2S+ ρSS− 2 N+1,j,k,j N+1,j k,j k,j N+1,j j,k X X M − γ0 {ρS,Sα Sα } − 2Sα ρSSα . N+1,j,k,j 2 N+1,j k,l k,l N+1,j α=± i,k,j6=l

For the energy shift term (the ﬁrst term) in expression (5.12), we have

+ − S − + SN+1,jSk.jρ = ρ1...(Sk.jρk)...ρN (SN+1,jρN+1) = 0 (4.18) S + − − + ρ SN+1,jSk.l = ρ1...(ρkSk.l)...ρN (ρN+1SN+1,j) = 0

65 For the thermal terms (the second and third terms) in expression (5.12), we have

S + − + − S ρ SN+1,jSk,j =SN+1,jSk,jρ = 0

S − + − + S ρ SN+1,jSk,j =SN+1,jSk,jρ = 0

− S + − + Sk,jρ SN+1,j =ρ1...(Sk.jρk)...ρN (ρN+1SN+1,j)

− + =ρ1...(Sk.1ρk)...ρN (ρN+1SN+1,1)

=ρ1...(A|bki)(Ahak| + Chck|)..ρN (4.19)

⊗ (A|aN+1i + C|cN+1i)(AhbN+1|)

+ S − + − Sk,jρ SN+1,j =ρ1...(Sk.jρk)...ρN (ρN+1SN+1,j)

+ − =ρ1...(Sk.2ρk)...ρN (ρN+1SN+1,2)

=ρ1...(C|bki)(Ahak| + Chck|)...ρN

⊗ (A|aN+1i + C|cN+1i)(ChbN+1|)

For the squeezed vacuum terms(the fourth term), we have

S α α α α S ρ SN+1,jSk,l = SN+1,jSk,lρ = 0

+ S + + + Sk,1ρ SN+1,2 = ρ1...(Sk,1ρk)...ρN (ρN+1SN+1,2) = 0

+ S + + + Sk,2ρ SN+1,1 = ρ1...(Sk,2ρk)...ρN (ρN+1SN+1,1)

= ρ1...(C|bki)(Ahak| + Chck|)...ρN

⊗ (A|aN+1i + C|cN+1i)(AhbN+1|), (4.20)

− S − − − Sk,1ρ SN+1,2 = ρ1...(Sk,1ρk)...ρN (ρN+1SN+1,2)

= ρ1...(A|bki)(Ahak| + Chck|)...ρN

⊗ (A|aN+1i + C|cN+1i)(ChbN+1|)

− S − − − Sk,2ρ SN+1,1 = ρ1...(Sk,2ρk)...ρN (ρN+1SN+1,1) = 0

66 On substituting from Eqs. (5.13)-(5.15) into expression (5.12), we have

X 2 − S + X 2 + S − γN+1,1,k,1(ch )Sk,1ρ SN+1,1 + γN+1,2,k,2sh Sk,2ρ SN+1,2 k j,k

X X 0 α S α + γN+1,j,k,lchshSk,lρ SN+1,j α=± i,k,j6=l

+ ρ1...(A|bki)(Ahak| + Chck|)...ρN (A|aN+1i + C|cN+1i)(ChbN+1|)]

X 2 2 2 2 0 = (γN+1,1,k,1ch A + γN+1,2,k,2sh C + 2γN+1,j,k,lchshCA) k

× ρ1...(|bki)(Ahak| + Chck|)..ρN (A|aN+1i + C|cN+1i)(hbN+1|).

2 2 2 2 0 It is not difﬁcutlt to prove that ch A γN+1,1,k,1 + sh C γN+1,2,k,2 + 2chshCAγN+1,j,k,l = 0 by substituting the expressions of A, B, C. Hence, the extra terms with the atom index i = N + 1 and k = 1 ∼ N when we add the N + 1th atom are 0. Similarly, the terms with i = 1 ∼ N and k = N + 1 also vanish. Thus, we prove that the right hand side of Eq. (4.16) is zero when the state of the system is direct product of the steady state of single atom. This indicates that direct product of the steady state of single atom is the steady state of the multiple atoms driven by the squeezed vacuum. It is interesting to note that while introducing the dipole-dipole interaction between the atoms affects the evolution of the system, the ﬁnal steady state still remains unaffected. Therefore, for multiple atoms, a population inversion of almost 100% can also be achieved even the dipole- dipole interaction is considered, under the condition that the dipole direction for all atoms are properly oriented to satisfy γab γbc. Actually, in the normal squeezed vacuum with correlation shown in Eq. (2.40), the steady state of the atoms can also be direct product of the steady state of a single atom if all the atoms are in the nodes of the standing wave. To verify the above proof, we will do the numerical simulation to show that the steady state

67 of multiple atoms is indeed the direct product of steady state of single atom. Since the cost for numerical simulation increases exponentially as the number of atoms increases, we only show the ﬁdelity of two-atom state with respect to theoretical steady state as a function of time in Fig. 4.6, where the system is initially in the ground state. From Fig. 4.6(a), we can see that different atom separations have different evolution dynamics because they have different dipole-dipole interactions. However, we can see that the system ﬁnally evolves into the following equation regardless of the atomic separation, squeezing parameter, and the ratio of decay rate γab/γbc: √ √ sh γ2 ch γ1 |ψssi = |a1i − |c1i pch2γ + sh2γ pch2γ + sh2γ 1 √ 2 1 √ 2 (4.22) sh γ2 ch γ1 ⊗ |a2i − |c2i p 2 2 p 2 2 ch γ1 + sh γ2 ch γ1 + sh γ2

From Fig. 4.6(b), we see that the system takes less time to evolve into the steady state for a smaller squeezing parameter. Fig. 4.6(c) shows that while smaller γab results in higher population inversion, it takes much longer for the system to evolve into the steady state.

68 5. CAVITY-CAVITY INTERACTION IN THE SQUEEZED VACUUM

In the previous sections, we studied the interactions between the squeezed vacuum and atoms, or fermions with discrete energy levels. In this section, we study the interactions between the squeezed vacuum and the harmonic oscillators, or bosons with continuous energy levels. A practi- cal model is a leaky cavity, where the modes inside and outside the cavity are essentially harmonic oscillators, and their coupling can be described by the Q factor[30].

5.1 General master equation of cavity-cavity interaction

In this section, we will derive the master equation for two single-mode leaky cavities placed inside the waveguide with the squeezed vacuum injected from both ends. The schematic setup is shown in Fig. 5.1. Then we will study how the modes inside the cavity will evolve under the inﬂuence of the squeezed vacuum. The free Hamiltonian of cavity and waveguide modes is:

X 1 X 1 H = ω (a†a + ) + ω (a† a + ) 0 ~ i i i 2 ~ k k,s k,s 2 (5.1) i k,s

where ak stands for the modes in the waveguide and ai is the ﬁeld operator of the single mode inside ith the cavity. The waveguide is saturated with the squeezed vacuum with the center frequency ω0. The interaction Hamiltonian between the cavity mode and waveguide modes is:

X + † V = −i~ [Daks − D aks] (5.2) ks where X ∗ † D = [gi,k,sai + gi,k,sai] (5.3) i

69 −ikzri Here we deﬁne gi,k,s = |gi,k,s|e where ri is just a phenomenological parameter describing the location of cavity. The reduced master equation of atoms in the reservoir is[30]

S Z t dρ 1 S F = − 2 dτT rF {[V (t), [V (t − τ), ρ (t − τ)ρ } dt ~ 0 Z t 1 S F S F (5.4) = − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ + ρ (t − τ)ρ V (t − τ)V (t) ~ 0 − V (t)ρS(t − τ)ρF V (t − τ) − V (t − τ)ρS(t − τ)ρF V (t)}.

Here we just show how to deal with the ﬁrst term in Eq.(5.4), the remaining terms can be calculated in the same way. For the ﬁrst term, we have

Z t 1 S F − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ } ~ 0 Z t X F + F † = dτ {D(t)D(t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] − D(t)D (t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] 0 ks,k0s0

+ F † + + F † † S − D (t)D(t − τ)T rF [ρ aks(t)ak0s0 (t − τ)] + D (t)D (t − τ)T rF [ρ aks(t)ak0s0 (t − τ)]}ρ (t − τ)}. (5.5)

Under the rotating wave approximation(RWA), we have

Z t 1 S F − 2 dτT rF {V (t)V (t − τ)ρ (t − τ)ρ } ~ 0 Z t X X ∗ † iωit ∗ † iωj (t−τ) −i(ωks+ω 0 0 )t+iω 0 0 τ k s k s 0 0 = dτ{gi,k,sai e gj,k0,s0 aje e [− sinh(r) cosh(r)δk ,2k0−kδss ] ij ks,k0s0 0

∗ † iωit −iωj (t−τ) −iωk0s0 τ 2 − gi,k,sai e gj,k0,s0 aje e cosh rδkk0 δss0

−iωit ∗ † iωj (t−τ) −iωk0s0 τ 2 − gi,k,saie gj,k0,s0 aje e cosh rδkk0 δss0

−iωit ∗ † iωj (t−τ) iωk0s0 τ 2 − gi,k,saie gj,k0,s0 aje e sinh rδkk0 δss0

∗ † iωit −iωj (t−τ) iωk0s0 τ 2 − gi,k,sai e gj,k0,s0 aje e sinh rδkk0 δss0

−iωit −iωj (t−τ) i(ωks+ω 0 0 )t−iω 0 0 τ S 0 0 k s k s 0 0 + gi,k,saie gj,k ,s aje e [− sinh(r) cosh(r)δk ,2k0−kδss ]}ρ (t − τ) (5.6)

70 Here we just calculate the ﬁrst and second term to show how to get the master equation. For the second term, we have

Z t X ∗ † iωit −iωj (t−τ) −iωk0s0 τ 2 S − dτgi,k,sai e gj,k0,s0 aje e cosh rρ (t − τ)δkk0 δss0 0 kz L Z ∞ Z t i(ωi−ωj )t iωj τ −iωk τ = − e dkz dτe e z |gi,k,sgj,k,s| 2π −∞ 0

ikz(ri−rj ) 2 † S × e cosh rai ajρ (t − τ) ∞ t L Z Z 2 i(ωi−ωj )t iωj τ −i[ωj +c kjz(kz−kjz)/ωj ]τ ≈ − e dkz dτe e |gi,k,sgj,k,s| 2π 0 0

ikz(ri−rj ) −ikz(ri−rj ) 2 † S × [e + e ] cosh rai ajρ (t − τ) ∞ t L Z Z 2 i(ωi−ωj )t −iτc kjzδkz/ωj ≈ − e dδkz dτe |gi,k,sgj,k,s| 2π −k0z 0

× [ei(kjz+δkz)(ri−rj ) + e−i(kjz+δkz)(ri−rj )] cosh2 ra†a ρS(t − τ) i j (5.7) ∞ t L Z Z 2 i(ωi−ωj )t −i(c kjzδkz/ωj )τ ≈ − e dδkz dτe |gi,k,sgj,k,s| 2π −∞ 0

i(kjz+δkz)(ri−rj ) −i(kjz+δkz)(ri−rj ) 2 † S × [e + e ] cosh rai ajρ (t − τ) L Z t c2k i(ωi−ωj )t ikjz(ri−rj ) jz ≈ − e dτ|gi,k,sgj,k,s|2π[e δ((ri − rj) − τ) 2π 0 ω0 c2k −ikjz(ri−rj ) jz 2 † S + e δ((ri − rj) + τ)] cosh rai ajρ (t − τ) ω0 L ωj ≈ − eikjzrij |g g |2π cosh2 ra†a ρS(t)ei(ωi−ωj )t 2π i,k,s j,k,s c2k i j √ √ 0z γiγj γiγj 2 † S i(ωi−ωj )t ≈ − [ cos(k0zrij) + i sin(k0zrij)] cosh rai ajρ (t)e √2 2 γiγj ≡ − ( + iΛ ) cosh2 ra†a ρS(t)ei(ωi−ωj )t 2 ij i j where rij = |ri −rj| is also a phenomenological parameter indicating the relative position between

2 √ cavities. γi = L|gi,k0 | is the leaking rate for the ith cavity, and Λij = γiγj sin(k0zrij)/2 is the

p π 2 2 energy shift. In the third line we expand ωk = c ( a ) + (kz) around kz = k0z since resonant R ∞ modes provide dominant contributions. In the ﬁfth line we extend the integration dkz → −k0z R ∞ −∞ dkz because the main contribution comes from the components around δkz = 0. In the next line, Weisskopf-Wigner approximation is used.

71 Next we need to calculate the ﬁrst term (squeezing term) in Eq.(5.6):

Z t i(ωi+ωj −2ω0)tX ∗ † ∗ † i(ωk−ωj )τ S e dτ{gi,2k0−kai gj,kaje [− sinh(r) cosh(r)]ρ (t − τ) 0 kz L Z 2k0z Z t i(ωi+ωj −2ω0)t i(ωk −ωj )τ i(2kiz−kz)(ri−o1) ikz(rj −o1) = − e dkz dτe z e e 2π 0 0 † † S (5.8) × |gi,2k0−kgj,k| sinh(r) cosh(r)ai ajρ (t − τ) L Z 0 Z t i(ωi+ωj −2ω0)t i(ωk −ωj )τ i(−2kiz−kz)(ri−o2) ikz(rj −o2) − e dkz dτe z e e 2π −2k0z 0

† † S × |gi,2k0−kgj,k| sinh(r) cosh(r)ai ajρ (t − τ)

Putting the overall factor ei(ωi+ωj −2ω0)t aside, for i = j, Eq.(5.8) reduces to

Z t X ∗ † ∗ † i(ωk−ωi)τ S dτ{gi,2k0−kai gi,kai e [− sinh(r) cosh(r)]ρ (t − τ) 0 kz 2 Z 2k0z Z t c kiz L i (kz−kiz)τ i2k (r −o ) † † S ωi 0z i 1 = − dkz dτe e |gi,2k0−kgi,k| sinh(r) cosh(r)ai ai ρ (t − τ) 2π 0 0 2 Z 0 Z t c kiz L i (kz−kiz)τ −i2k (r −o ) † † S ωi 0z i 2 − dkz dτe e |gi,2k0−kgi,k| sinh(r) cosh(r)ai ai ρ (t − τ) 2π −2k0z 0 L Z t c2k i2k0z(ri−o1) −i2k0z(ri−o2) iz † † S = − [e + e ]|gi,2k0−kgi,k| dτ2πδ( τ) sinh(r) cosh(r)ai ai ρ (t − τ) 2π 0 ωi γi = −ei2k0zR cos(2k r ) sinh(r) cosh(r)a†a†ρS(t) 2 0z i i i (5.9) where we have used the fact that the origin of coordinate system is at equal distance from two

0 sources(i.e., o2 = −o1 = R) in the second last line. Thus, we have γii = γi cos(2k0zri). For

72 ri 6= rj, Eq. (5.8) reduces to

Z t X † † i(ωk−ωj )τ S dτ{gi,2k0−kai gj,kaje [− sinh(r) cosh(r)]ρ (t − τ) 0 kz 2 Z 2k0z Z t c kjz L i (kz−kjz)τ ω i2k0z(rc−o1) −i(kz−k0z)(ri−rj ) = − dkz dτe j e e 2π 0 0 † † S × |gi,2k0−kgj,k| sinh(r) cosh(r)ai ajρ (t − τ) 2 Z 0 Z t c kjz L i (−kz−kjz)τ ω −i2k0z(rc−o2) −i(kz+k0z)(ri−rj ) − dkz dτe j e e 2π −2k0z 0

i(kz−k0z)(ri−rj ) † † S × e sinh(r) cosh(r)ai ajρ (t − τ) L Z t c2k i2k0zR i2k0zrc 0z ≈ − e |gi,2k0−kgj,k| dτ2π[e δ(ri − rj − τ) 2π 0 ω0 c2k −i2k0zrc 0z † † S + e δ(ri − rj + τ)] sinh(r) cosh(r)ai ajρ (t − τ) ω0

i2k0zR i2k0zrcsgn(i−j) † † S ≈ − e L|gi,k0 gj,k0 |e ai ajρ (t) √ γiγj → − ei2k0zR cos(k (r + r ))a†a†ρS(t) 2 0z i j i j where sgn(i − j) is the sign function. The last arrow is because we need to sum over i, j, so the

i2k0zrcsgn(i−j) 0 i2k0zR√ imaginary part of e vanishes and the neat result is that γij = e γiγj cos(k0z(ri +

† S † rj)). As for ai ρ (t)aj terms, the combination of the last two terms in Eq.(5.4) will make the

i2k0zrcsgn(i−j) 0 i2k0zR√ imaginary part of e vanish. Thus, we have γij = e γiγj cos(k0z(ri + rj)). Doing the above calculation for all terms in Eq.(5.4), we have the general equation for cavity-

73 Figure 5.1: (a) Schematic setup: two single-mode cavities are placed inside the waveguide with the broadband squeezed vacuum incident from both ends.

cavity interaction in the squeezed vacuum as follows:

X 2 † † † i(ωi−ωj )t ρ˙ = γij cosh r(−ρai aj − ai ajρ + 2aiρaj)e ij

X 2 † † † −i(ωi−ωj )t + γij sinh r(−ρaiaj − aiajρ + 2ai ρaj)e (5.11) ij

X iθ iθ iθ −i(ωi+ωj )t + γij cosh r sinh r[(e ρaiaj + e aiajρ − e 2aiρaj)e + H.c.] ij

5.2 Steady state of non-resonant cavities

First, we study two non-resonant cavities coupled to the squeezed vacuum reservoir. The eigen frequencies of these two cavities are ω1 = ω0 − δω and ω2 = ω0 + δω. Under the rotating wave approximation(RWA), Eq.(5.11) becomes:

X † † † ρ˙ = γ(1 + N)(−ρai ai − ai aiρ + 2aiρai ) i

X † † † + γN(−ρaiai − aiai ρ + 2ai ρai) (5.12) i

X iθ iθ iθ + γM(e ρaiaj + e aiajρ − 2e aiρaj + h.c.) i6=j

74 where we have assumed γij = γ for simplicity. The above equation can be re-arranged as:

X γ ρ˙ = [−ρ(cosh(r)a† − eiθ sinh(r)a )(cosh(r)a − e−iθ sinh(r)a†) 2 i j i j i6=j

† iθ −iθ † (5.13) − (cosh(r)ai − e sinh(r)aj)(cosh(r)ai − e sinh(r)aj)ρ

−iθ † † iθ + 2(cosh(r)ai − e sinh(r)aj)ρ(cosh(r)ai − e sinh(r)aj)]

we use the following Bogoliubov transformation[58]:

? † † S = exp(η aiaj − ηai aj)

+ −iθ † Ai = S aiS = cosh(r)ai − e sinh(r)aj (5.14)

+ + + + iθ Ai = S ai S = cosh(r)ai − e sinh(r)aj

so the master equation Eq.(5.13) becomes:

X † † † ρ˙ = γ[−ρAi Ai − Ai Aiρ + 2AiρAi ] (5.15) i

† Next we redeﬁne the density matrix: ρs = SρS . Thus Eq.(5.15) becomes:

X † † † ρ˙s = γ[−ρsai ai − ai aiρs + 2aiρsai ] i (5.16) X l† l r† r r l† ≡ γ[−ai aiρs − ai ai ρs + 2ai ai ρs] ≡ Lρs i

l l† r l† Here we deﬁne superoperator {ai, ai }({ai , ar } ) only acting to the left(right) on density operator ρ [59, 60]. These operators have the following commutation relations:

r r† l l† l r† l r l† r l† r† [ai , aj ] = δij, [ai, aj ] = −δij, [ai, aj ] = [ai, aj ] = [ai , aj ] = [ai , aj ] = 0 (5.17)

Thus, the steady state of Eq.(5.16) can be solved by solving Lρ = 0, which requires the diagno-

−aral†−aral† lization of superoperator L. Applying the similarity transformation U = e 1 1 2 2 to Eq.(5.16)

75 −1 r† l r l† r† l† r l r l† , since we have U (ai , ai, ai , ai )U = (ai + ai , ai + ai, ai , ai ), the right hand side of Eq.(5.16) becomes:

X −1 l† l r† r r l† −1 X l† l r† r −1 RHS = γU [−ai ai − ai ai + 2ai ai ]UU ρs = γ[−ai ai − ai ai ]U ρs (5.18) i i

−1 † † −K−1−K−2 The only solution to Lρ = 0 is U ρs = |0, 0ih0, 0|, which yields ρ = S ρSS = S e |0, 0ih0, 0|S = S†|0, 0ih0, 0|S which is the two mode squeezed vacuum.

5.3 Steady state of resonant cavities

Next we study the case where two cavities are identical, i.e., ω1 = ω2 = ω0. Then the master equation becomes:

X 2 † † † ρ˙ = γ cosh r(−ρai aj − ai ajρ + 2ajρai ) ij

X 2 † † † + γ sinh r(−ρaiaj − aiajρ + 2ajρai) (5.19) ij

X iθ iθ iθ + γ cosh r sinh r(e ρaiaj + e aiajρ − e 2aiρaj + h.c.) ij

This equation can be rearranged as follows:

X † iθ −iθ † ρ˙ = γ[−ρ(cosh rai − e sinh rai)(cosh raj − e sinh raj) ij

† iθ −iθ † (5.20) − (cosh rai − e sinh rai)(cosh raj − e sinh raj)ρ

−iθ † † iθ + 2(cosh raj − e sinh raj)ρ(cosh rai − e sinh rai)]

We introduce the Bogoliubov transformation:

1 1 S = exp( η?a2 − ηa†2) i 2 i 2 i + −iθ † (5.21) Ai = Si aiSi = cosh(r)ai − e sinh(r)ai

+ + + + iθ Ai = Si ai Si = cosh(r)ai − e sinh(r)ai

76 so master equation Eq.(5.20) becomes

X † † † ρ˙ = γ[−ρAi Aj − Ai Ajρ + 2AjρAi ] (5.22) ij

+˙ + Next we deﬁne ρs = S1S2ρS1 S2 so the master equation is reduced to:

X † † † ρ˙s = γ[−ρsai aj − ai ajρs + 2ajρsai ] (5.23) ij

To diagnolize this Lindblad equation, we introduce the transformation:

1 L1 √ (a1 − a2) = 2 L √1 (a + a ) 2 2 1 2

† where [Li,Lj] = δij, and the master equation becomes:

† † † ρ˙s = γ[−2ρsL2L2 − 2L2L2ρs + 4L2ρsL2]

r† r l† l l r† = γ[−2L2 L2ρs − 2L2 L2ρs + 4L2L2 ρs] (5.24)

= Lρ

† Operator L2 has the following properties:

† 1 L |0i = √ (|01i + |10i) ≡ |1L i 2 2 2

√ √ √ † 1 1 L √ (|01i + |10i) = 2[ (|02i + 2|11i + |20i)] = 2|2L i 2 2 2 2 √ √ √ √ √ † 1 1 L (|02i + 2|11i + |20i) = 3[ √ (|03i + 3|12i + 3|21i + |30i)] = 3|3L i 2 2 2 2 2

...

77 † while the operator L1 has the following properties:

† 1 L |0i = √ (−|01i + |10i) ≡ |1L i 1 2 1

√ √ √ † 1 1 L √ (−|01i + |10i) = 2[ (|02i − 2|11i + |20i)] = 2|2L i 1 2 2 1 √ √ √ √ √ † 1 1 L (|02i − 2|11i + |20i) = 3[ √ (−|03i + 3|12i − 3|21i + |30i)] = 3|3L i 1 2 2 2 1

...

Thus, L1 and L2 is just another representation of a1 and a2. Then we use the similarity transfor-

r l† r† l† r† l† l† −L2L2 −1 l r l r r mation: U = e , which yields U (L2 ,L2,L2 ,L2)U = (L2 + L2 ,L2 + L2,L2 ,L2). Thus, the master equation Eq.(5.25) becomes:

−1 l† l r† r r l† −1 l† l r† r −1 RHS = γU [−L2 L2 − L2 L2 + 2L2L2 ]UU ρs = γ[−L2 L2 − L2 L2]U ρs (5.25)

r l† −L2L2 The solutions to the steady state are ρs = e |0L2 mL1 ih0L2 nL1 | = |mL1 ihnL1 | which yields a†−a† ρ = S+S+ √1 ( 1√ 2 )m|0ih0| √1 ( a1√−a2 )nS S . This solution degenerates to the single mode 1 2 m! 2 n! 2 1 2 P squzeed vacuum in two modes when m = n = 0. Generally, an initial state ρ(0) = mnpq Cmnpq|mnihpq| = P 0 P P 0 mnpq Cmnpq|mL1 pL2 ihnL1 qL2 | will evolve into mn Gmn|mL1 ihnL1 | where Gmn = mnp Cmnpp. Therefore, we have shown that the entangled modes in the squeezed vacuum can be physically separated by the resonant cavities, without any loss of entanglement between them.

78 6. SUMMARY AND CONCLUSIONS1

In this dissertation, we systematically studied the interactions between the squeezed vacuum and the atoms. We challenged the traditional reservoir theory which fails to consider the effect of the squeezing source. We put forward a new reservoir theory by modifying the mode function of the electromagnetic fields, which includes the position information of the squeezing source. Then we derived a master equation of the atomic dynamics based on the Weisskopf-Wigner approximation. In our formalism, the density matrix is naturally positive-definite. We then apply this theory to the 1D waveguide-QED system where the squeezing in one direction is experimentally achievable. We show that the enhancement and suppression of the dephasing rate caused by the squeezed vacuum is actually dependent its position in the waveguide. In single-atom case, the squeezing does not affect its population dynamics. However, in multi-atom case, the squeezing can strongly affect the population dynamics of the system because two-photon absorption and emission are allowed in multi-atom system. We also show that dipole-dipole interaction influences dephasing rate and we can tune the position of the squeezing source to tune the dephasing rate of the system. Moreover, we show that stationary entangled state can be achieved in this system independent of the initial state and the emitter separation. Particularly, when the center of mass is close to nλ0z/4 and the squeezing is large, the system can be prepared in GHZ state. Moreover, we study the power spectrum of the resonance fluorescence. It is demonstrated that the phase of the squeezed vacuum, emitter separation, and the center-of-mass position can affect the bandwidth and the intensity of the sidebands. We further generalized our theory to arbitrary atomic structures. We studied the Ξ-type atoms coupled to a broadband squeezed vacuum reservoir in a quasi-one-dimensional waveguide, with

the overall transition frequency ωac = 2ω0. We showed that a single atom evolves into a steady

79 state which is a superposition of the second excited state and the ground state. If the decay rate from the second excited state to the first excited state is much smaller than that from the first excited state to the ground state, the population can be almost 100% trapped in the second excited state, which is a great improvement compared to the maximum ratio of 78% in Ref. [39]. What is more, we proved that the above result can be generalized to an arbitrary number of atoms interacting with each other via dipole-dipole interaction, and the system’s final steady state is a direct product of that in the single-atom case with modified squeezed vacuum shown in Eq. (4.14). This is one of the most interesting results here and its physical insight still needs further studies. We also argued that the arbitrary ratio of the two transitions’ decay rates can be effectively controlled by different waveguide structure. This population-inversed system is experimentally feasible since the experiments on the broadband squeezed vacuum coupled to the artificial atom in a 1D cavity have been widely conducted[26, 28, 29, 61, 62, 63].

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